91Ó°ÊÓ

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? $$ \sum_{n=0}^{\infty} \frac{(2 x+3)^{2 n+1}}{n !} $$

Find the binomial series for the functions in Exercises \(11-14\) . $$ \left(1-\frac{x}{2}\right)^{4} $$

In Exercises \(13-22,\) find a formula for the \(n\) th term of the sequence. The sequence \(-1,1,-1,1,-1, \ldots\)

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

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? $$ \sum_{n=0}^{\infty} \frac{x^{n}}{\sqrt{n^{2}+3}} $$

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

Find the Maclaurin series for the functions in Exercises \(9-20 .\) $$ 7 \cos (-x) $$

Use partial fractions to find the sum of each series. $$ \sum_{n=1}^{\infty} \frac{4}{(4 n-3)(4 n+1)} $$

Which of the series in Exercises \(11-44\) converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{n}{n^{3}+1} $$

Find Taylor series at \(x=0\) for the functions in Exercises \(7-18\) $$ \frac{x^{2}}{1-2 x} $$