91Ó°ÊÓ

Which of the series in Exercises \(1-26\) converge, and which diverge? Give reasons for your answers. (When checking your answers, remember there may be more than one way to determine a series' convergence or divergence.) $$ \sum_{n=1}^{\infty} \frac{3^{n}}{n^{3} 2^{n}} $$

Which of the series in Exercises 1–36 converge, and which diverge? Give reasons for your answers. $$ \sum_{n=1}^{\infty} \frac{10 n+1}{n(n+1)(n+2)} $$

Find series solutions for the initial value problems in Exercises \(15-32\) . $$ y^{\prime \prime}+y=x, \quad y^{\prime}(0)=1 \text { and } y(0)=2 $$

Converge, and which diverge? Give reasons for your answers. If a series converges, find its sum. $$ \sum_{n=0}^{\infty} \cos n \pi $$

For what positive values of \(x\) can you replace \(\ln (1+x)\) by \(x\) with an error of magnitude no greater than 1\(\%\) of the value of \(x ?\)

In Exercises \(1-32,\) (a) find the series' radius and interval of convergence. For what values of \(x\) does the series converge (b) absolutely, (c) conditionally? $$ \sum_{n=2}^{\infty} \frac{x^{n}}{n(\ln n)^{2}} $$

Which of the series converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.) $$ \sum_{n=1}^{\infty} \frac{8 \tan ^{-1} n}{1+n^{2}} $$

Which of the series in Exercises \(11-44\) converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ \sum_{n=1}^{\infty}(-1)^{n} \frac{n}{n+1} $$

Which of the sequences \(\left\\{a_{n}\right\\}\) in Exercises \(23-84\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\frac{1-5 n^{4}}{n^{4}+8 n^{3}} $$

In Exercises \(21-28\) , find the Taylor series generated by \(f\) at \(x=a\) $$ f(x)=e^{x}, \quad a=2 $$