Convergence of Series and Sequences.

[DaManWitDaPlan]

Hey, guys. I have a Calculus II test in 7 hours, and I need some help...

The following is my review. Any help is greatly appreciated.
(an is a series variable for each question)

Math 227-011 Fall 2009 Test 3 ___________________________________
True/False:
____ 1. If a sequence is bounded, it must converge.
____ 2. If a sequence converges, it must be bounded.
____ 3. If the terms of a series approach zero, then the series must converge.
____4. The sequence whose nth term is
1+ 1/n
converges.
____5. If
(sum from 1 to inf)an
converges and
0 < an < bn for all n,
then
(sum)bn
must converge.


Examples:
Give an example of the following or explain why no such example exists:
6. a power series with interval of convergence (5, 7).
7. a power series which converges at x = 1 and at x = 3 but diverges at x = 2.
8. an alternating series which does not converge absolutely.
9. a power series
(sum from 0 to inf)(an * x^n)
which converges on (-2, 2) but whose derivative only
converges on (-1,1).
10. two divergent series such that the (sum from 1 to inf) of an + bn converges.
11. Give the first four terms of the sequence of partial sums of the series (sum from 1 to inf)(2^(-2n))


This is just a little of it, but if I can get help solving any of them, I will be very happy

 

[galactus]

11. Give the first four terms of the sequence of partial sums of the series \(\displaystyle \sum_{n=1}^{\infty}2^{-2n}=\sum_{n=1}^{\infty}\frac{1}{4^{n}}\)

Use the geometric series. \(\displaystyle \frac{\frac{1}{4}}{1-\frac{1}{4}}\)

That is the sum of the series. Actually, it can be shown that \(\displaystyle \sum_{n=1}^{\infty}\frac{1}{a^{n}}=\frac{1}{a-1}, \;\ a>1\)