Calculus II · Unit 4A · lesson

Alternating-Series Error Estimates

Certify alternating-series approximations with the first omitted term and solve for the smallest number of terms meeting a tolerance.

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Section overview

Alternating, absolute, ratio, and root tests

What this section is building

Certify alternating-series approximations with the first omitted term and solve for the smallest number of terms meeting a tolerance.

Notice

Absolute convergence controls magnitude strongly enough to imply convergence; conditional convergence relies on cancellation.

Decide

Test absolute values first when practical, use the alternating-series hypotheses explicitly, and reserve ratio or root tests for matching algebraic structure.

Avoid

Calling any alternating-looking series convergent or treating a ratio/root limit of one as a verdict.

Concept

Learning objectives

use the first omitted term to control approximation error for a convergent alternating series.

Alternating-Series Error Estimates

Explanation

The next term measures the remaining uncertainty

For a decreasing alternating series, successive partial sums lie on opposite sides of the limit. The true sum is trapped between two consecutive partial sums, so the distance from either one to the limit cannot exceed the next term magnitude. This unusually clean estimate makes alternating series computationally attractive.

The bound is a guarantee, not an exact error. To meet a tolerance, solve $b_{N+1}<\varepsilon$ . Then check indexing carefully: the first omitted term after $s_N$ is term $N+1$ . If the series begins at another index, translate the term count rather than mechanically substituting $N+1$ .

Bridge

The next term controls the entire remaining error

For an alternating series satisfying the test, the true sum lies between every pair of consecutive partial sums. The error after $N$ terms is therefore no larger than the distance to the next endpoint of that bracket, namely $b_{N+1}$ .

This unusually simple bound turns convergence into a practical approximation method. To guarantee a tolerance, solve $b_{N+1}<\varepsilon$ . The sign of the first omitted term also tells which side of the true sum the approximation lies on.

Proof idea

The true sum stays inside the next bracket

The alternating partial sums nest around $S$ . Since $S$ lies between $s_N$ and $s_{N+1}$ , its distance from $s_N$ is at most $|s_{N+1}-s_N|=b_{N+1}$ .

Concept

Alternating remainder estimate

Under the Alternating Series Test hypotheses,

|S-s_N|\le b_{N+1}.

Guided walkthrough

Approximate $\ln2$

Using

\ln2=1-\frac12+\frac13-\cdots,

we want error below $0.01$ . Since the next term after $N$ terms has magnitude $1/(N+1)$ , require $1/(N+1)<0.01$ , so $N\ge100$ guarantees the tolerance.

Worked example

Guarantee four-decimal accuracy

For

\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^3},

how many terms guarantee error below $10^{-4}$ ? The alternating remainder satisfies

|R_N|\le \frac{1}{(N+1)^3}.

Require $(N+1)^3>10^4$ . Since $22^3=10648$ , $N+1=22$ works, so $N=21$ terms guarantee the requested accuracy.

Common mistake

Use the first omitted term

After summing through $n=N$ , the bound is $b_{N+1}$ , not $b_N$ . This one-index error can invalidate an accuracy claim.

Interactive checku4a-alternating_series_error-01

What bounds the error after $N$ terms of the alternating harmonic series?

Your work stays on this device. No account or AI grader is used.

Show hint

Use the magnitude of the first omitted term.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Exercise

How many terms guarantee error below $10^{-3}$ ?

Exercise

Approximate $\sum(-1)^{n-1}/n^2$ with four terms and state an error bound.

Exercise

Explain why the true sum lies between consecutive partial sums.

Exercise

Compare this error estimate with the integral remainder estimate.

Source & rights

Original instruction with traceable references.

BetterGrades-original; no direct adaptation declared in the verified handoff.

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.

Alternating-Series Error Estimates

Explanation

The next term measures the remaining uncertainty

Bridge

The next term controls the entire remaining error

Proof idea

The true sum stays inside the next bracket

The alternating partial sums nest around $S$ . Since $S$ lies between $s_N$ and $s_{N+1}$ , its distance from $s_N$ is at most $|s_{N+1}-s_N|=b_{N+1}$ .

Concept

Alternating remainder estimate

Under the Alternating Series Test hypotheses,

|S-s_N|\le b_{N+1}.

Guided walkthrough

Approximate $\ln2$

Using

\ln2=1-\frac12+\frac13-\cdots,

we want error below $0.01$ . Since the next term after $N$ terms has magnitude $1/(N+1)$ , require $1/(N+1)<0.01$ , so $N\ge100$ guarantees the tolerance.

Worked example

Guarantee four-decimal accuracy

For

\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^3},

how many terms guarantee error below $10^{-4}$ ? The alternating remainder satisfies

|R_N|\le \frac{1}{(N+1)^3}.

Require $(N+1)^3>10^4$ . Since $22^3=10648$ , $N+1=22$ works, so $N=21$ terms guarantee the requested accuracy.

Common mistake

Use the first omitted term

After summing through $n=N$ , the bound is $b_{N+1}$ , not $b_N$ . This one-index error can invalidate an accuracy claim.

Interactive checku4a-alternating_series_error-01

What bounds the error after $N$ terms of the alternating harmonic series?

Your work stays on this device. No account or AI grader is used.

Show hint

Use the magnitude of the first omitted term.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Exercise

How many terms guarantee error below $10^{-3}$ ?

Exercise

Approximate $\sum(-1)^{n-1}/n^2$ with four terms and state an error bound.

Exercise

Explain why the true sum lies between consecutive partial sums.

Exercise

Compare this error estimate with the integral remainder estimate.

Source & rights

Original instruction with traceable references.

BetterGrades-original; no direct adaptation declared in the verified handoff.

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.

Alternating-Series Error Estimates

Related resources

Original instruction with traceable references.

Alternating-Series Error Estimates

Related resources

Original instruction with traceable references.