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Calculus II · Unit 4B · reference

Unit 4B Reference Sheet

Unit 4B Reference Sheet

11−x=∑n=0∞xn,∣x∣<1\frac1{1-x}=\sum_{n=0}^\infty x^n,\quad |x|<1ex=∑n=0∞xnn!,sin⁡x=∑n=0∞(−1)nx2n+1(2n+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=∑n=0∞(−1)nx2n(2n)!,ln⁡(1+x)=∑n=1∞(−1)n−1xnn.\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}.Pn(x)=∑k=0nf(k)(a)k!(x−a)k,∣Rn(x)∣≤M∣x−a∣n+1(n+1)!.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)!}.

After the explanation

Use the section idea

Reading lens

Mixed work tests recognition: classify the task before selecting a convergence, construction, transformation, or error method.

Mental model

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

Decision

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

Common trap

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

Check yourself

Can you reproduce the method and domain without the key?

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Original instruction with traceable references.

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