Question

Please, help
Determine X_n converges or diverges
\(X_n=\dfrac{(-1)^n*n}{n+1}\)

Answer 1

I have solution here. What I don't know is how to link \(X_n\) and \(X_{n+2} \) to complete the proof.

Answer 2

@ganeshie8

Answer 3

Clearly the two subsequences are convergent

Answer 4

since they are converging to different values, the overall sequence wont converge

Answer 5

as n goes to infinity,
the even terms are converging to 1
and the odd terms are converging to -1

Answer 6

oh, use convergent subsequences.

Answer 7

so you will see the terms as :
...., 0.98, -0.98, 0.99, -0.99,...

Answer 8

so the soverall sequence is dancing between -1 and 1forever

Answer 9

Can I use limit theorem to show it diverges?
For this proof, they use subsequence, but the problem asks me to use limit theorem, which is : if Xn = Yn *Zn, and BOTH Yn, Zn converge, then Xn converges
I have Yn =(-1)^n diverges
Zn = \(\dfrac{n}{n+1}\) converges to 1 ( I can prove it)
But I don't know whether I can apply that theorem to state that Xn diverges

Answer 10

yeah its one way i think :
if Yn and Zn converge, then Yn*Zn converges

Answer 11

if Yn diverges and Zn converges, then we `don't know` if Yn*Zn diverges or not.

Answer 12

Got you. Thank you very much. I make a lot of questions to make sure that I understand the concept completely. :)
Thanks for being patient to me. :)

Answer 13

thats good only :)

here is an example :

Yn = n diverges
Zn = 1/n^2 converges

but the sequence Yn*Zn = 1/n converges.

Answer 14

:)