Determine if the following Serie converges or not

$$\Large{\sum_{k\in\mathbb{N}}\frac{(-1)^{k}}{k^{2}+1}}$$

Question

Determine if the following Serie converges or not

$$\Large{\sum_{k\in\mathbb{N}}\frac{(-1)^{k}}{k^{2}+1}}$$

anonymous · Answer

Alternating series test.

anonymous · Answer

i know but i couldnt apply it.. can u show me please ? i have exam on wendensday  =(

anonymous · Answer

Well, show the absolute value is always decreasing.

anonymous · Answer

i think the test is for showing if the serie monoton decreased.. bu

anonymous · Answer

hmm how to apply it?

anonymous · Answer

Show $a_n > a_{n+1}$.

anonymous · Answer

ok i will make a step but i would be happy if you help me by the next steps..

anonymous · Answer

Well, the absolute value is always decreasing.

anonymous · Answer

$$ n\frac{1}{(n+1)^2+1}\ |a_n|>|a_{n+1}| $$

anonymous · Answer

$$\Large{\sum_{k\in\mathbb{N}}\frac{(-1)^{k}}{k^{2}+1}=(-1)^{k}(\frac{1}{k^{2}+1})}$$ and i think $$\Large{ \frac{1}{k^{2}+1}\geq \frac{1}{(k+1)^{2}+1}}$$

anonymous · Answer

aah ok.. adn how to plugin it to question so that in exam i can get full points for this question?

anonymous · Answer

Just invoke the alternating series test.

anonymous · Answer

ok i will try..

anonymous · Answer

thank you very much