convergent test of series

[fjalhada]

Does this series converge or diverge?

I need the proof.

 

[BigBeachBanana]

Use limit test. \(\displaystyle L=\lim_{n\to \infty}a_n\\\)
If [imath]|L|>0[/imath] or [imath]L[/imath] is undefined then the series diverges.

 

[pka]

Does this series converge or diverge?
View attachment 31568
I need the proof.

SEE THIS As you see [imath]\mathop {\lim }\limits_{n \to \infty } \left( {1 - \cos \left( {\frac{{n - 1}}{{{n^2}}}} \right)} \right) = 0 [/imath] it passes the first test, so it may converge. i.e. it is not necessarily divergent.
This is a complicated argument: SEE HERE