91Ó°ÊÓ

If \(\sum C_{n}(x-3)^{n}\) converges at \(x=7\) and diverges at \(x=10,\) what can you say about the convergence at \(x=11 ?\) At \(x=5 ?\) At \(x=0 ?\)

The series \(\sum a_{n}\) converges. Explain, by looking at partial sums, why the series \(\sum\left(a_{n+1}-a_{n}\right)\) also converges.

Show that the sequence \(s_{n}\) satisfies the recurrence relation. $$\begin{aligned} &s_{n}=2 n^{2}-n\\\ &s_{n}=s_{n-1}+4 n-3 \text { for } n>1 \text { and } s_{1}=1 \end{aligned}$$

For the given value of \(x\) determine whether the infinite geometric series converges. If so, find its sum: $$3+3 \cos x+3(\cos x)^{2}+3(\cos x)^{3}+\cdots$$ $$x=0$$

(a) Decide whether the following series is alternating: $$\sum_{n=1}^{\infty} \frac{\sin n}{n^{3}}$$ (b) Use the comparison test to determine whether the following series converges or diverges: $$\sum_{n=1}^{\infty}\left|\frac{\sin n}{n^{3}}\right|$$ (c) Determine whether the following series converges or diverges: $$\sum_{n=1}^{\infty} \frac{\sin n}{n^{3}}$$

The series \(\sum C_{n} x^{n}\) converges at \(x=-5\) and diverges at \(x=7 .\) What can you say about its radius of convergence?

Use the formula for \(s_{n}\) to give the third term of the sequence, \(s_{3}\) $$s_{n}=(-1)^{n} 2^{n-1} \cdot n^{2}$$

The series \(\sum a_{n}\) diverges. Give examples that show the series \(\sum\left(a_{n+1}-a_{n}\right)\) could converge or diverge.

For the given value of \(x\) determine whether the infinite geometric series converges. If so, find its sum: $$3+3 \cos x+3(\cos x)^{2}+3(\cos x)^{3}+\cdots$$ $$x=2 \pi / 3$$

The series \(\sum C_{n}(x+7)^{n}\) converges at \(x=0\) and diverges at \(x=-17 .\) What can you say about its radius of convergence?