91Ó°ÊÓ

Find a good bound for the maximum value of the given expression, given that \(c\) is in the stated interval. Answers may vary depending on the technique used. $$ |\tan c+\sec c| ;\left[0, \frac{\pi}{4}\right] $$

Indicate whether the given series converges or diverges and give a reason for your conclusion. $$\sum_{n=2}^{\infty}\left(1-\frac{1}{n}\right)^{n}$$

Suppose that \(f(x)=\sum_{n=0}^{\infty} a_{n} x^{n}=\sum_{n=0}^{\infty} b_{n} x^{n}\) for \(|x|

Let \(k\) be an arbitrary number and \(-1

Classify each series as absolutely convergent, conditionally convergent, or divergent. \(\sum_{n=1}^{\infty}(-1)^{n+1} \sin \frac{\pi}{n}\)

By writing \(1 / x=1 /[1-(1-x)]\) and using the known expansion of \(1 /(1-x),\) find the Taylor series for \(1 / x\) in powers of \(x-1\).

Find a good bound for the maximum value of the given expression, given that \(c\) is in the stated interval. Answers may vary depending on the technique used. $$ \left|\frac{4 c}{\sin c}\right| ;\left[\frac{\pi}{4}, \frac{\pi}{2}\right] $$

Indicate whether the given series converges or diverges and give a reason for your conclusion. $$\sum_{n=1}^{\infty} n^{2}\left(\frac{2}{3}\right)^{n}$$

Find the power series representation of \(x /\left(x^{2}-3 x+2\right)\). Hint: Use partial fractions.

In Problems \(31-36,\) write the first four terms of the sequence \(\left\\{a_{n}\right\\}\). Then use Theorem \(D\) to show that the sequence converges. $$ a_{n}=\frac{4 n-3}{2^{n}} $$