Question

Determine if the following Serie converges or not

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

Answer 1

Alternating series test.

Answer 2

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

Answer 3

Well, show the absolute value is always decreasing.

Answer 4

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

Answer 5

hmm how to apply it?

Answer 6

Show \(a_n > a_{n+1}\).

Answer 7

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

Answer 8

Well, the absolute value is always decreasing.

Answer 9

\[
n<n+1\\
n^2<(n+1)^2\\
n^2+1<(n+1)^2+1\\
\frac{1}{n^2+1}>\frac{1}{(n+1)^2+1}\\
|a_n|>|a_{n+1}|
\]

Answer 10

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

Answer 11

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

Answer 12

Just invoke the alternating series test.

Answer 13

ok i will try..

Answer 14

thank you very much