Calculus II · Unit 4B · reference

Unit 4B Reference Sheet

Use Unit 4B Reference Sheet as a concise reference for formulas, theorem hypotheses, decisions, and common errors.

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Section overview

Review, practice, exams, and answer keys

What this section is building

Use Unit 4B Reference Sheet as a concise reference for formulas, theorem hypotheses, decisions, and common errors.

Notice

A complete solution preserves the series identity, interval, endpoint logic, and approximation claim as one chain.

Decide

Write the objective and conditions first, solve independently, then use one answer reveal to repair the earliest incorrect decision.

Avoid

Checking algebra while leaving the convergence domain or error certification unstated.

Unit 4B Reference Sheet

\frac1{1-x}=\sum_{n=0}^\infty x^n,\quad |x|<1

e^x=\sum_{n=0}^\infty\frac{x^n}{n!},\quad \sin x=\sum_{n=0}^\infty(-1)^n\frac{x^{2n+1}}{(2n+1)!},

\cos x=\sum_{n=0}^\infty(-1)^n\frac{x^{2n}}{(2n)!}, \quad \ln(1+x)=\sum_{n=1}^\infty(-1)^{n-1}\frac{x^n}{n}.

P_n(x)=\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a)^k, \qquad |R_n(x)|\le\frac{M|x-a|^{n+1}}{(n+1)!}.

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.

Unit 4B Reference Sheet

\frac1{1-x}=\sum_{n=0}^\infty x^n,\quad |x|<1

e^x=\sum_{n=0}^\infty\frac{x^n}{n!},\quad \sin x=\sum_{n=0}^\infty(-1)^n\frac{x^{2n+1}}{(2n+1)!},

\cos x=\sum_{n=0}^\infty(-1)^n\frac{x^{2n}}{(2n)!}, \quad \ln(1+x)=\sum_{n=1}^\infty(-1)^{n-1}\frac{x^n}{n}.

P_n(x)=\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a)^k, \qquad |R_n(x)|\le\frac{M|x-a|^{n+1}}{(n+1)!}.

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.