Is This Infinite Series Convergent?

[Imaxx]

Homework Statement

solve this infinite series

Relevant Equations

.

∞
∑ (n∧2+3n+1) / (n∧4+2n∧3+n∧2) =?
n=1

I attempted to find the general sum of this 'expression'?? But no luck. How can I solve this?

 

[fresh_42]

Do you know what $\sum n$ and $\sum n^2$ are?

 

[Imaxx]

from what to what? 1 to ∞?

 

[fresh_42]

from what to what? 1 to ∞?

Sorry, I missed the denominator. You should really have used LaTeX. One formula isn't a big deal.

Can you cancel your quotient and factorize the denominator?

 

[fresh_42]

If you have no typo, then you will probably have to consider the sequence of partial sums.

 

[vela]

You might try separating the original series into three series, one for each term in the numerator, and seeing if you can evaluate those.

Just do a partial fraction expansion. It's a telescoping series.

 

[archaic]

https://www.physicsforums.com/help/latexhelp/

$$\sum_{n=1}^{\infty} \frac{n^2+3n+1}n^2(n+1)^2}$$

$$\sum_{n=1}^{\infty} \frac{n^2+3n+1}{n^2(n+1)^2}$$

 

[archaic]

I feel a bit ashamed that there's a mistake in the code block after getting two likes.
Here's the correct version

$$\sum_{n=1}^{\infty} \frac{n^2+3n+1}{n^2(n+1)^2}$$

Sorry for bumping up the post.

 

[etotheipi]

I feel a bit ashamed that there's a mistake in the code block after getting two likes.
Here's the correct version

$$\sum_{n=1}^{\infty} \frac{n^2+3n+1}{n^2(n+1)^2}$$

Sorry for bumping up the post.

Now I feel ashamed for not noticing either

 

[Dr Transport]

Try breaking the fraction into partial fractions

 

[fresh_42]

The OP cheated and managed to inveigle his way to a full solution:
https://www.physicsforums.com/threads/how-to-prove-this-infinite-series.992669/

I like to take the chance and remind you of our rules:

• please report homework questions in technical forums, instead of answering them
• do not provide full answers, that doesn't help the OP to understand their problem, even in technical forums
• do not open multiple threads on the same topic
• homework questions (anyway where they have been posted) require some efforts to be shown from the OP. We are not a solution automaton. Our goal is to teach, not to solve.

This thread is closed.