Worksheet · Calculus II · Unit 4B

Taylor Series Worksheet with Answers and Error Practice

Twenty problems on known series, transformations, coefficients, intervals, approximations, and error.

20 problems75 min estimated timeIntermediate to advanced progression

What is included

Practice Maclaurin and Taylor series with complete solutions, interval checks, and error estimates.

Skills assessed

series construction
known-series transformations
intervals of convergence
remainder estimates

Prerequisites

derivatives
infinite series

Taylor Series Worksheet instructional sequence — Twenty problems on known series, transformations, coefficients, intervals, approximations, and error. The numbered labels and written sequence preserve meaning without relying on color.

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Find the degree-3 Maclaurin polynomial for e^x.
Find the degree-5 Maclaurin polynomial for sin x.
Find the degree-4 Maclaurin polynomial for cos x.
Write the first four nonzero terms of ln(1+x).
Expand 1/(1-x) as a power series.
Expand 1/(1+x^2) as a power series.
Expand 1/(1-3x) as a power series.
Find a power series for 1/(1-x)^2.
Find a power series for arctan x by integrating 1/(1+x²).
Find the degree-2 Taylor polynomial for ln x centered at 1.
Write the Taylor series for e^x centered at a.
If f^(n)(0)=2^n, find the Maclaurin series.
Use T_3 for e^x to approximate e^{0.1}.
Use x-x³/6 to approximate sin(0.2).
Find the radius of $\sum n x^n/4^n$ .
Find the interval of convergence of $\sum x^n/n$ .
Find the interval of convergence of $\sum (x-2)^n/(n3^n)$ .
Find R for $\sum n!(x+1)^n/(2n)!$ .
Bound the error of the degree-3 Maclaurin approximation to e^{0.2}.
How many terms of the alternating arctan series ensure error below 0.001 at x=1/2?

Complete worked solutions

Every problem has a source-matched answer and independently reviewed derivation.

Problem 1: Find the degree-3 Maclaurin polynomial for e^x.

Answer: $1+x+x^2/2+x^3/6$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Start from derivatives at the center or a known convergent power series.
Apply substitution, differentiation, integration, or the ratio test while preserving the convergence condition.
The resulting expression is $1+x+x^2/2+x^3/6$ .

Problem 2: Find the degree-5 Maclaurin polynomial for sin x.

Answer: $x-x^3/6+x^5/120$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Start from derivatives at the center or a known convergent power series.
Apply substitution, differentiation, integration, or the ratio test while preserving the convergence condition.
The resulting expression is $x-x^3/6+x^5/120$ .

Problem 3: Find the degree-4 Maclaurin polynomial for cos x.

Answer: $1-x^2/2+x^4/24$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Start from derivatives at the center or a known convergent power series.
Apply substitution, differentiation, integration, or the ratio test while preserving the convergence condition.
The resulting expression is $1-x^2/2+x^4/24$ .

Problem 4: Write the first four nonzero terms of ln(1+x).

Answer: $x-x^2/2+x^3/3-x^4/4$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Start from derivatives at the center or a known convergent power series.
Apply substitution, differentiation, integration, or the ratio test while preserving the convergence condition.
The resulting expression is $x-x^2/2+x^3/3-x^4/4$ .

Problem 5: Expand 1/(1-x) as a power series.

Answer: $\sum_{n=0}^{\infty}x^n,\ |x|<1$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Start from derivatives at the center or a known convergent power series.
Apply substitution, differentiation, integration, or the ratio test while preserving the convergence condition.
The resulting expression is $\sum_{n=0}^{\infty}x^n,\ |x|<1$ .

Problem 6: Expand 1/(1+x^2) as a power series.

Answer: $\sum_{n=0}^{\infty}(-1)^nx^{2n},\ |x|<1$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Start from derivatives at the center or a known convergent power series.
Apply substitution, differentiation, integration, or the ratio test while preserving the convergence condition.
The resulting expression is $\sum_{n=0}^{\infty}(-1)^nx^{2n},\ |x|<1$ .

Problem 7: Expand 1/(1-3x) as a power series.

Answer: $\sum_{n=0}^{\infty}3^nx^n,\ |x|<1/3$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Start from derivatives at the center or a known convergent power series.
Apply substitution, differentiation, integration, or the ratio test while preserving the convergence condition.
The resulting expression is $\sum_{n=0}^{\infty}3^nx^n,\ |x|<1/3$ .

Problem 8: Find a power series for 1/(1-x)^2.

Answer: $\sum_{n=1}^{\infty}nx^{n-1},\ |x|<1$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Start from derivatives at the center or a known convergent power series.
Apply substitution, differentiation, integration, or the ratio test while preserving the convergence condition.
The resulting expression is $\sum_{n=1}^{\infty}nx^{n-1},\ |x|<1$ .

Problem 9: Find a power series for arctan x by integrating 1/(1+x²).

Answer: $\sum_{n=0}^{\infty}(-1)^n x^{2n+1}/(2n+1)$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Start from derivatives at the center or a known convergent power series.
Apply substitution, differentiation, integration, or the ratio test while preserving the convergence condition.
The resulting expression is $\sum_{n=0}^{\infty}(-1)^n x^{2n+1}/(2n+1)$ .

Problem 10: Find the degree-2 Taylor polynomial for ln x centered at 1.

Answer: $(x-1)-(x-1)^2/2$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Start from derivatives at the center or a known convergent power series.
Apply substitution, differentiation, integration, or the ratio test while preserving the convergence condition.
The resulting expression is $(x-1)-(x-1)^2/2$ .

Problem 11: Write the Taylor series for e^x centered at a.

Answer: $e^a\sum_{n=0}^{\infty}(x-a)^n/n!$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Start from derivatives at the center or a known convergent power series.
Apply substitution, differentiation, integration, or the ratio test while preserving the convergence condition.
The resulting expression is $e^a\sum_{n=0}^{\infty}(x-a)^n/n!$ .

Problem 12: If f^(n)(0)=2^n, find the Maclaurin series.

Answer: $\sum_{n=0}^{\infty}2^nx^n/n!=e^{2x}$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Start from derivatives at the center or a known convergent power series.
Apply substitution, differentiation, integration, or the ratio test while preserving the convergence condition.
The resulting expression is $\sum_{n=0}^{\infty}2^nx^n/n!=e^{2x}$ .

Problem 13: Use T_3 for e^x to approximate e^{0.1}.

Answer: $1.105166\overline6$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Start from derivatives at the center or a known convergent power series.
Apply substitution, differentiation, integration, or the ratio test while preserving the convergence condition.
The resulting expression is $1.105166\overline6$ .

Problem 14: Use x-x³/6 to approximate sin(0.2).

Answer: $0.198666\overline6$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Start from derivatives at the center or a known convergent power series.
Apply substitution, differentiation, integration, or the ratio test while preserving the convergence condition.
The resulting expression is $0.198666\overline6$ .

Problem 15: Find the radius of $\sum n x^n/4^n$ .

Answer: $R=4$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Start from derivatives at the center or a known convergent power series.
Apply substitution, differentiation, integration, or the ratio test while preserving the convergence condition.
The resulting expression is $R=4$ .

Problem 16: Find the interval of convergence of $\sum x^n/n$ .

Answer: $[-1,1)$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Start from derivatives at the center or a known convergent power series.
Apply substitution, differentiation, integration, or the ratio test while preserving the convergence condition.
The resulting expression is $[-1,1)$ .

Problem 17: Find the interval of convergence of $\sum (x-2)^n/(n3^n)$ .

Answer: $[-1,5)$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Start from derivatives at the center or a known convergent power series.
Apply substitution, differentiation, integration, or the ratio test while preserving the convergence condition.
The resulting expression is $[-1,5)$ .

Problem 18: Find R for $\sum n!(x+1)^n/(2n)!$ .

Answer: $R=\infty$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Start from derivatives at the center or a known convergent power series.
Apply substitution, differentiation, integration, or the ratio test while preserving the convergence condition.
The resulting expression is $R=\infty$ .

Problem 19: Bound the error of the degree-3 Maclaurin approximation to e^{0.2}.

Answer: $|R_3|\le e^{0.2}(0.2)^4/4!<0.000082$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Start from derivatives at the center or a known convergent power series.
Apply substitution, differentiation, integration, or the ratio test while preserving the convergence condition.
The resulting expression is $|R_3|\le e^{0.2}(0.2)^4/4!<0.000082$ .

Problem 20: How many terms of the alternating arctan series ensure error below 0.001 at x=1/2?

Answer: $4\text{ terms}$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Start from derivatives at the center or a known convergent power series.
Apply substitution, differentiation, integration, or the ratio test while preserving the convergence condition.
The resulting expression is $4\text{ terms}$ .

Common errors

Dropping factorials in coefficient formulas.
Transforming a series without its convergence interval.
Using an error estimate without checking its hypotheses.

Worksheet · Calculus II · Unit 4B

Taylor Series Worksheet with Answers and Error Practice

Twenty problems on known series, transformations, coefficients, intervals, approximations, and error.

20 problems75 min estimated timeIntermediate to advanced progression

What is included

Practice Maclaurin and Taylor series with complete solutions, interval checks, and error estimates.

Skills assessed

series construction
known-series transformations
intervals of convergence
remainder estimates

Prerequisites

derivatives
infinite series

Printable and accessible

Download this resource

No email address or account is required.

Student PDF Worked answer key PDF SVG PNG

Device-only study control

Ready for an honest first attempt?

This control sends only the resource id and type—never your answers or identity.

Printable preview

Find the degree-3 Maclaurin polynomial for e^x.
Find the degree-5 Maclaurin polynomial for sin x.
Find the degree-4 Maclaurin polynomial for cos x.
Write the first four nonzero terms of ln(1+x).
Expand 1/(1-x) as a power series.
Expand 1/(1+x^2) as a power series.
Expand 1/(1-3x) as a power series.
Find a power series for 1/(1-x)^2.
Find a power series for arctan x by integrating 1/(1+x²).
Find the degree-2 Taylor polynomial for ln x centered at 1.
Write the Taylor series for e^x centered at a.
If f^(n)(0)=2^n, find the Maclaurin series.
Use T_3 for e^x to approximate e^{0.1}.
Use x-x³/6 to approximate sin(0.2).
Find the radius of $\sum n x^n/4^n$ .
Find the interval of convergence of $\sum x^n/n$ .
Find the interval of convergence of $\sum (x-2)^n/(n3^n)$ .
Find R for $\sum n!(x+1)^n/(2n)!$ .
Bound the error of the degree-3 Maclaurin approximation to e^{0.2}.
How many terms of the alternating arctan series ensure error below 0.001 at x=1/2?

Complete worked solutions

Every problem has a source-matched answer and independently reviewed derivation.

Problem 1: Find the degree-3 Maclaurin polynomial for e^x.

Answer: $1+x+x^2/2+x^3/6$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Start from derivatives at the center or a known convergent power series.
Apply substitution, differentiation, integration, or the ratio test while preserving the convergence condition.
The resulting expression is $1+x+x^2/2+x^3/6$ .

Problem 2: Find the degree-5 Maclaurin polynomial for sin x.

Answer: $x-x^3/6+x^5/120$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Start from derivatives at the center or a known convergent power series.
Apply substitution, differentiation, integration, or the ratio test while preserving the convergence condition.
The resulting expression is $x-x^3/6+x^5/120$ .

Problem 3: Find the degree-4 Maclaurin polynomial for cos x.

Answer: $1-x^2/2+x^4/24$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Start from derivatives at the center or a known convergent power series.
Apply substitution, differentiation, integration, or the ratio test while preserving the convergence condition.
The resulting expression is $1-x^2/2+x^4/24$ .

Problem 4: Write the first four nonzero terms of ln(1+x).

Answer: $x-x^2/2+x^3/3-x^4/4$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Start from derivatives at the center or a known convergent power series.
Apply substitution, differentiation, integration, or the ratio test while preserving the convergence condition.
The resulting expression is $x-x^2/2+x^3/3-x^4/4$ .

Problem 5: Expand 1/(1-x) as a power series.

Answer: $\sum_{n=0}^{\infty}x^n,\ |x|<1$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Start from derivatives at the center or a known convergent power series.
Apply substitution, differentiation, integration, or the ratio test while preserving the convergence condition.
The resulting expression is $\sum_{n=0}^{\infty}x^n,\ |x|<1$ .

Problem 6: Expand 1/(1+x^2) as a power series.

Answer: $\sum_{n=0}^{\infty}(-1)^nx^{2n},\ |x|<1$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Start from derivatives at the center or a known convergent power series.
Apply substitution, differentiation, integration, or the ratio test while preserving the convergence condition.
The resulting expression is $\sum_{n=0}^{\infty}(-1)^nx^{2n},\ |x|<1$ .

Problem 7: Expand 1/(1-3x) as a power series.

Answer: $\sum_{n=0}^{\infty}3^nx^n,\ |x|<1/3$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Start from derivatives at the center or a known convergent power series.
Apply substitution, differentiation, integration, or the ratio test while preserving the convergence condition.
The resulting expression is $\sum_{n=0}^{\infty}3^nx^n,\ |x|<1/3$ .

Problem 8: Find a power series for 1/(1-x)^2.

Answer: $\sum_{n=1}^{\infty}nx^{n-1},\ |x|<1$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Start from derivatives at the center or a known convergent power series.
Apply substitution, differentiation, integration, or the ratio test while preserving the convergence condition.
The resulting expression is $\sum_{n=1}^{\infty}nx^{n-1},\ |x|<1$ .

Problem 9: Find a power series for arctan x by integrating 1/(1+x²).

Answer: $\sum_{n=0}^{\infty}(-1)^n x^{2n+1}/(2n+1)$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Start from derivatives at the center or a known convergent power series.
Apply substitution, differentiation, integration, or the ratio test while preserving the convergence condition.
The resulting expression is $\sum_{n=0}^{\infty}(-1)^n x^{2n+1}/(2n+1)$ .

Problem 10: Find the degree-2 Taylor polynomial for ln x centered at 1.

Answer: $(x-1)-(x-1)^2/2$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Start from derivatives at the center or a known convergent power series.
Apply substitution, differentiation, integration, or the ratio test while preserving the convergence condition.
The resulting expression is $(x-1)-(x-1)^2/2$ .

Problem 11: Write the Taylor series for e^x centered at a.

Answer: $e^a\sum_{n=0}^{\infty}(x-a)^n/n!$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Start from derivatives at the center or a known convergent power series.
Apply substitution, differentiation, integration, or the ratio test while preserving the convergence condition.
The resulting expression is $e^a\sum_{n=0}^{\infty}(x-a)^n/n!$ .

Problem 12: If f^(n)(0)=2^n, find the Maclaurin series.

Answer: $\sum_{n=0}^{\infty}2^nx^n/n!=e^{2x}$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Start from derivatives at the center or a known convergent power series.
Apply substitution, differentiation, integration, or the ratio test while preserving the convergence condition.
The resulting expression is $\sum_{n=0}^{\infty}2^nx^n/n!=e^{2x}$ .

Problem 13: Use T_3 for e^x to approximate e^{0.1}.

Answer: $1.105166\overline6$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Start from derivatives at the center or a known convergent power series.
Apply substitution, differentiation, integration, or the ratio test while preserving the convergence condition.
The resulting expression is $1.105166\overline6$ .

Problem 14: Use x-x³/6 to approximate sin(0.2).

Answer: $0.198666\overline6$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Start from derivatives at the center or a known convergent power series.
Apply substitution, differentiation, integration, or the ratio test while preserving the convergence condition.
The resulting expression is $0.198666\overline6$ .

Problem 15: Find the radius of $\sum n x^n/4^n$ .

Answer: $R=4$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Start from derivatives at the center or a known convergent power series.
Apply substitution, differentiation, integration, or the ratio test while preserving the convergence condition.
The resulting expression is $R=4$ .

Problem 16: Find the interval of convergence of $\sum x^n/n$ .

Answer: $[-1,1)$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Start from derivatives at the center or a known convergent power series.
Apply substitution, differentiation, integration, or the ratio test while preserving the convergence condition.
The resulting expression is $[-1,1)$ .

Problem 17: Find the interval of convergence of $\sum (x-2)^n/(n3^n)$ .

Answer: $[-1,5)$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Start from derivatives at the center or a known convergent power series.
Apply substitution, differentiation, integration, or the ratio test while preserving the convergence condition.
The resulting expression is $[-1,5)$ .

Problem 18: Find R for $\sum n!(x+1)^n/(2n)!$ .

Answer: $R=\infty$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Start from derivatives at the center or a known convergent power series.
Apply substitution, differentiation, integration, or the ratio test while preserving the convergence condition.
The resulting expression is $R=\infty$ .

Problem 19: Bound the error of the degree-3 Maclaurin approximation to e^{0.2}.

Answer: $|R_3|\le e^{0.2}(0.2)^4/4!<0.000082$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Start from derivatives at the center or a known convergent power series.
Apply substitution, differentiation, integration, or the ratio test while preserving the convergence condition.
The resulting expression is $|R_3|\le e^{0.2}(0.2)^4/4!<0.000082$ .

Problem 20: How many terms of the alternating arctan series ensure error below 0.001 at x=1/2?

Answer: $4\text{ terms}$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Start from derivatives at the center or a known convergent power series.
Apply substitution, differentiation, integration, or the ratio test while preserving the convergence condition.
The resulting expression is $4\text{ terms}$ .

Common errors

Dropping factorials in coefficient formulas.
Transforming a series without its convergence interval.
Using an error estimate without checking its hypotheses.

What is included

Skills assessed

Prerequisites

Download this resource

Ready for an honest first attempt?

Printable preview

Complete worked solutions

Problem 1: Find the degree-3 Maclaurin polynomial for e^x.

Problem 2: Find the degree-5 Maclaurin polynomial for sin x.

Problem 3: Find the degree-4 Maclaurin polynomial for cos x.

Problem 4: Write the first four nonzero terms of ln(1+x).

Problem 5: Expand 1/(1-x) as a power series.

Problem 6: Expand 1/(1+x^2) as a power series.

Problem 7: Expand 1/(1-3x) as a power series.

Problem 8: Find a power series for 1/(1-x)^2.

Problem 9: Find a power series for arctan x by integrating 1/(1+x²).

Problem 10: Find the degree-2 Taylor polynomial for ln x centered at 1.

Problem 11: Write the Taylor series for e^x centered at a.

Problem 12: If f^(n)(0)=2^n, find the Maclaurin series.

Problem 13: Use T_3 for e^x to approximate e^{0.1}.

Problem 14: Use x-x³/6 to approximate sin(0.2).

Problem 15: Find the radius of ∑nxn/4n\sum n x^n/4^n∑nxn/4n.

Problem 16: Find the interval of convergence of ∑xn/n\sum x^n/n∑xn/n.

Problem 17: Find the interval of convergence of ∑(x−2)n/(n3n)\sum (x-2)^n/(n3^n)∑(x−2)n/(n3n).

Problem 18: Find R for ∑n!(x+1)n/(2n)!\sum n!(x+1)^n/(2n)!∑n!(x+1)n/(2n)!.

Problem 19: Bound the error of the degree-3 Maclaurin approximation to e^{0.2}.

Problem 20: How many terms of the alternating arctan series ensure error below 0.001 at x=1/2?

Common errors

Continue learning

Lessons and articles

Related resources

Glossary

What is included

Skills assessed

Prerequisites

Download this resource

Ready for an honest first attempt?

Printable preview

Complete worked solutions

Problem 1: Find the degree-3 Maclaurin polynomial for e^x.

Problem 2: Find the degree-5 Maclaurin polynomial for sin x.

Problem 3: Find the degree-4 Maclaurin polynomial for cos x.

Problem 4: Write the first four nonzero terms of ln(1+x).

Problem 5: Expand 1/(1-x) as a power series.

Problem 6: Expand 1/(1+x^2) as a power series.

Problem 7: Expand 1/(1-3x) as a power series.

Problem 8: Find a power series for 1/(1-x)^2.

Problem 9: Find a power series for arctan x by integrating 1/(1+x²).

Problem 10: Find the degree-2 Taylor polynomial for ln x centered at 1.

Problem 11: Write the Taylor series for e^x centered at a.

Problem 12: If f^(n)(0)=2^n, find the Maclaurin series.

Problem 13: Use T_3 for e^x to approximate e^{0.1}.

Problem 14: Use x-x³/6 to approximate sin(0.2).

Problem 15: Find the radius of ∑nxn/4n\sum n x^n/4^n∑nxn/4n.

Problem 16: Find the interval of convergence of ∑xn/n\sum x^n/n∑xn/n.

Problem 17: Find the interval of convergence of ∑(x−2)n/(n3n)\sum (x-2)^n/(n3^n)∑(x−2)n/(n3n).

Problem 18: Find R for ∑n!(x+1)n/(2n)!\sum n!(x+1)^n/(2n)!∑n!(x+1)n/(2n)!.

Problem 19: Bound the error of the degree-3 Maclaurin approximation to e^{0.2}.

Problem 20: How many terms of the alternating arctan series ensure error below 0.001 at x=1/2?

Common errors

Continue learning

Lessons and articles

Related resources

Glossary

Problem 15: Find the radius of $\sum n x^n/4^n$ .

Problem 16: Find the interval of convergence of $\sum x^n/n$ .

Problem 17: Find the interval of convergence of $\sum (x-2)^n/(n3^n)$ .

Problem 18: Find R for $\sum n!(x+1)^n/(2n)!$ .

Problem 15: Find the radius of $\sum n x^n/4^n$ .

Problem 16: Find the interval of convergence of $\sum x^n/n$ .

Problem 17: Find the interval of convergence of $\sum (x-2)^n/(n3^n)$ .

Problem 18: Find R for $\sum n!(x+1)^n/(2n)!$ .