Series and convergence/divergence help

[Gauss177]

Hey all, I am really struggling to understand this chapter about series. These are a few problems about convergence and divergence, and I'll probably have some questions about Taylor and maclaurin series when I do those problems too.

Homework Statement

1. Determine whether the sequence is convergent or divergent. If it is convergent, find its limit.

a_n = 9^(n+1) / 10^n

Determine whether the series is convergent or divergent:
2. 1 / n(ln n)^2, series starts at n=2 and goes to infinity

3. Find the sum of the series:
[arctan (n+1) - arctan n], series starts at n=1 and goes to infninity.

Homework Equations

The Attempt at a Solution

1. I broke the problem down to: 9(9/10)^n, and said that as n->infinity the sequence also goes to infinity, so it's divergent. Need my method checked on that one.

2. Don't know about this one

3. Examples I've seen of these kinds of problems end up being a geometric series, so I just use a/1-r to find out sum of the series. But I don't think this applies here, so what else should I do?

Thanks for any help

 

[quasar987]

1. I broke the problem down to: 9(9/10)^n, and said that as n->infinity the sequence also goes to infinity, so it's divergent. Need my method checked on that one.

Are you sure about this. For instance, the first two terms are 9*9/10=8.1, 9*81/100=7.29. The sequence seems to be decreasing. Indeed, if you have a number 'a' such that -1<a<1, then the result of raising 'a' to any power greater than one results in a number lesser than 'a'.

I would be surprised if you were asked this question and nowhere in your textbook or class notes it was mentioned that the limit of a^n is 0 for any a such that -1<a<1.

 

[sutupidmath]

1.the first series is pretty easy, it converges and its limit is 0.
evaluate it using dallamber's test, or criterion. DO you know what it states?
2. for the second one i did not solve it all, but i think that considering that the series 1/ln*n diverges, than i think that considering this you can find if your serie converges or diverges! So i think the comparison test will work on second one. As for the third i have no idea.

 

[Dick]

For the third one, just write out the first few terms and scratch your head.

 

[Dick]

And the second one needs an integral test.