91Ó°ÊÓ

Which series in Exercises \(49-68\) converge, and which diverge? Give reasons for your answers. If a series converges, find its sum. $$ \sum_{n=0}^{\infty} e^{-2 n} $$

Determine how many terms should be used to estimate the sum of the entire series with an error of less than \(0.001 .\) $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{(n+3 \sqrt{n})^{3}} $$

Convergence or Divergence Which of the series in Exercises \(55-62\) converge, and which diverge? Give reasons for your answers. $$\sum_{n=1}^{\infty} \frac{2^{n} n ! n !}{(2 n) !}$$

The series $$e^{x}=1+x+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\frac{x^{4}}{4 !}+\frac{x^{5}}{5 !}+\cdots$$ converges to \(e^{x}\) for all \(x .\) a. Find a series for \((d / d x) e^{x} .\) Do you get the series for \(e^{x} ?\) Explain your answer. b. Find a series for \(\int e^{x} d x .\) Do you get the series for \(e^{x} ?\) Explain your answer. c. Replace \(x\) by \(-x\) in the series for \(e^{x}\) to find a series that con- verges to \(e^{-x}\) for all \(x .\) Then multiply the series for \(e^{x}\) and \(e^{-x}\) to find the first six terms of a series for \(e^{-x} \cdot e^{x} .\)

Convergence or Divergence Which of the series in Exercises \(55-62\) converge, and which diverge? Give reasons for your answers. $$\sum_{n=1}^{\infty} \frac{(-1)^{n}(3 n) !}{n !(n+1) !(n+2) !}$$

Show that the Taylor series for \(f(x)=\tan ^{-1} x\) diverges for \(|x|>1 .\)

Which of the sequences \(\left\\{a_{n}\right\\}\) in Exercises \(27-90\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\sqrt[n]{n^{2}} $$

Determine how many terms should be used to estimate the sum of the entire series with an error of less than \(0.001 .\) $$ \sum_{n=1}^{\infty}(-1)^{n} \frac{1}{\ln (\ln (n+2))} $$

If \(\sum_{n=1}^{\infty} a_{n}\) is a convergent series of nonnegative numbers, can anything be said about \(\sum_{n=1}^{\infty}\left(a_{n} / n\right) ?\) Explain.

Convergence or Divergence Which of the series in Exercises \(55-62\) converge, and which diverge? Give reasons for your answers. $$\sum_{n=1}^{\infty} \frac{(n !)^{n}}{\left(n^{n}\right)^{2}}$$