Does series sum[n=0,infty] [(n^3 cos(n pi)) / (n^4+3n^2+1)] converge cond'ly, conv. abs'ly, or diverge?

[Sophdof1]

I thought about using the alternating series theorem... but I’m unsure if there might be an easier way.

Also, can someone check if, what I have written is correct as I am pretty confident it is.

Thanks for your time

 

[Sophdof1]

I thought of using the alternating series test?

 

[pka]

I thought of using the alternating series test?

Can you show that the non-alternating part is decreasing and has a limit of zero?
The last part should be quite easy. So what about the decreasing part. Hint: I would look at the derivative.

 

[Sophdof1]

Can you show that the non-alternating part is decreasing and has a limit of zero?
The last part should be quite easy. So what about the decreasing part. Hint: I would look at the derivative.

I see the non alternating terms decrease, however the issue I am having is.. if you start at n=0 then this is not the case as it goes 0, 1/4, 8/41 etc ?

 

[Sophdof1]

I see the non alternating terms decrease, however the issue I am having is.. if you start at n=0 then this is not the case as it goes 0, 1/4, 8/41 etc ?

Could I do this?

 

[pka]

There is an old saying: All that matters happens in the tail of the series.

 

[Sophdof1]

There is an old saying: All that matters happens in the tail of the series.

I am now trying to show that the derivative of associated function is negative for x>1.. but I’m having troubles proving the actual inequality ..

 

[pka]

I am now trying to show that the derivative of associated function is negative for x>1.. but I’m having troubles proving the actual inequality ..

LOOK AT THIS What does tell you about the sequence from \(\displaystyle n=2\) on?

 

[Sophdof1]

LOOK AT THIS What does tell you about the sequence from \(\displaystyle n=2\) on?

It is decreasing ?

 

[pka]

It is decreasing ?

Bingo! We have a decreasing null-sequence. Are those the necessary conditions for convergent using the alternating series test ?