Sum of a Convergent Series: sum [n = 1, infty] [(2 + n) / (1 - 2n)]

[crybloodwing]

So I have the following.

I am a little confused on how to find the sum.

I know if it was in the form arⁿ​ then I could use a/1-r to figure the sum out.

 

[Dr.Peterson]

So I have the following.

I am a little confused on how to find the sum.

I know if it was in the form arⁿ​ then I could use a/1-r to figure the sum out.

The series is \(\displaystyle \displaystyle \sum_{n=1}^{\infty}\frac{2+n}{1-2n}\).

What is the condition for a series to be convergent? It is not just that the terms must have a limit ...

If I wanted to go further with the summation, I would use long division to rewrite the fraction as a "mixed expression" with a simpler fraction part.

 

[tkhunny]

So I have the following.

I am a little confused on how to find the sum.

I know if it was in the form arⁿ​ then I could use a/1-r to figure the sum out.

If you are tying to find "the sum" of this expression, you are definitely confused.

#1 - Prove that it converges at all. You seem to think it is a Geometric Series. Is it?
#1 - Prove that it converges at all. First requirement for an infinite sum is vanishing terms (approach a limit of zero(0)). Do they?

Without that, there is no point to trying to find a sum.

#2 - Under what circumstances does it converge?