# Infinite series/sequences

Hi I have a few problems with regard to infinite series and sequences:

1.) cos(n) / (n^2) summation from n = 0 to infinity

2.) cos(pi*n)/n summation from n = 0 to infinity

3.) sin(n) summation from 0 to infinity

I have to tell whether they absolutely converge, conditionally converge, or diverge.

I tried using the ration/root/limit test for them but nothing seemed to help.

any feedback will be greatly appreciated.

morphism
Homework Helper
The "comparison test" and the "divergence test" will be useful. For the second one, in particular, knowing what cos(pi*n) looks like is also helpful.

By the way, for the first two the summation can't possibly start at n=0. mathwonk
Homework Helper
2020 Award
there are two basic limits we an compute, geometric series, and improper integrals. all other series must be compared usually to one of these. a necessary condition is that the terms go to zero, which is sufficient if their size goes to zero monotonically and they have alternating signs.

compare 1 above to the series 1/n^2 which is then compoared to the integral of 1/x^2.

2 is the other test , alternating test.

3 i am not sure of. but suspect it does not converge by the test of whether the terms approach zero.

Gib Z
Homework Helper
My brain is exploding.

True, the series diverges by several tests, most obviously by the n-term test mathwonk mentioned. But if we go about evaluating the result;

$$S = \sum_{n=1}^{\infty} \sin (n) = \sum_{n=1}^{\infty} \mathRR{Im} (e^{in}) = \mathRR{Im}\sum_{n=1}^{\infty}(e^{in}) = \mathRR{Im} (\frac{1}{1-e^i}) = \frac{1+ \cos 1}{2 \sin^2 1}$$.

What did i do wrong :(

The geometric series doesn't converge when |z|=1:

$$s_n = \sum_{i=0}^n z^n = \frac{1}{1-z} - \frac{z^{n+1}}{1-z}$$, z not equal 1.

It's bounded, but is not a Cauchy sequence when |z| = 1.
($$s_{n+1}-s_n = e^{i \theta n}$$).

But as mentioned above, the main point is that the terms z^n don't go to zero as n -> infty when |z| = 1.

$$\sum c_n$$ converges $\Rightarrow$ $$s_n$$ is Cauchy $\Rightarrow$ $$|c_n| = |s_n-s_{n-1}| \rightarrow 0$$.

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