91Ó°ÊÓ

Write out the first eight terms of each series to show how the series starts. Then find the sum of the series or show that it diverges. $$\sum_{n=0}^{\infty}\left(\frac{5}{2^{n}}-\frac{1}{3^{n}}\right)$$

(a) find the series' radius and interval of convergence. For what values of \(x\) does the series converge (b) absolutely,(c) conditionally? $$\sum_{n=0}^{\infty} \frac{3^{n} x^{n}}{n !}$$

Find the binomial series for the functions. $$\left(1+x^{2}\right)^{3}$$

Each of Exercises gives the first term or two of a sequence along with a recursion formula for the remaining terms. Write out the first ten terms of the sequence. $$a_{1}=2, \quad a_{2}=-1, \quad a_{n+2}=a_{n+1} / a_{n}$$

Use the Root Test to determine whether each series converges absolutely or diverges. $$\sum_{n=1}^{\infty}\left(-\ln \left(e^{2}+\frac{1}{n}\right)\right)^{n+1}$$

Use the Integral Test to determine whether the series converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied. $$\sum_{n=2}^{\infty} \frac{1}{5 n+10 \sqrt{n}}$$

Find the binomial series for the functions. $$(1-2 x)^{3}$$

Find a formula for the \(n\) th term of the sequence. $$1,-1,1,-1,1, \ldots . \quad \text { I's with alternating signs }$$

Write out the first eight terms of each series to show how the series starts. Then find the sum of the series or show that it diverges. $$\sum_{n=0}^{\infty}\left(\frac{1}{2^{n}}+\frac{(-1)^{n}}{5^{n}}\right)$$

Use power series operations to find the Taylor series at \(x=0\) for the functions. $$x e^{x}$$