91Ó°ÊÓ

Suppose that \(\sum a_{n}\left(a_{n} \neq 0\right)\) is convergent. Prove that \(\sum 1 / a_{n}\) is divergent.

Use the Squeeze Theorem for Sequences to prove that $$ \lim _{n \rightarrow \infty} \sqrt[n]{a}=1 \quad a>0 $$ Hint: For \(n\) sufficiently large, \(1 / n

Suppose that \(\sum a_{n}\) is convergent and \(\sum b_{n}\) is divergent. Prove that \(\Sigma\left(a_{n}+b_{n}\right)\) is divergent. Hint: Prove by contradiction, using Theorem \(4 .\)

Formula (5) in Table 1 can be used to compute the value of \(\ln x\) for \(-1

Suppose that \(\sum a_{n}\) is divergent and \(c \neq 0\). Prove that \(\Sigma c a_{n}\) is divergent. Hint: Prove by contradiction, using Theorem \(4 .\)

Let \(f\) be the function defined by $$ f(x)=\left\\{\begin{array}{ll} e^{-1 / x^{2}} & \text { if } x \neq 0 \\ 0 & \text { if } x=0 \end{array}\right. $$ Show that \(f\) cannot be represented by a Maclaurin series.

Prove Theorem 1: If \(\lim _{x \rightarrow \infty} f(x)=L\) and \(\left\\{a_{n}\right\\}\) is a sequence defined by \(a_{n}=f(n)\), where \(n\) is a positive integer, then \(\lim _{n \rightarrow \infty} a_{n}=L\).

Prove Theorem 4: If \(\lim _{n \rightarrow \infty}\left|a_{n}\right|=0\), then \(\lim _{n \rightarrow \infty} a_{n}=0\).

Prove that if the sequence \(\left\\{a_{n}\right\\}\) converges, then the series \(\Sigma\left(a_{n+1}-a_{n}\right)\) converges. Conversely, prove that if \(\Sigma\left(a_{n+1}-a_{n}\right)\) converges, then \(\left\\{a_{n}\right\\}\) converges.

a. Find the Taylor series for \(f(x)=2 x^{3}+3 x^{2}+1\) at \(x=1\) b. Show that the Taylor series and \(f(x)\) are equal. c. What can you say about a Taylor series for a polynomial function? Justify your answer.