Question

Is there an easier way to solve the problem 2n+1/((n^2)((n+1)^2)) than adding up the sum from n to infinity?

Answer 1

I know the answer is 1, n = 1 as well. My problem is a more efficient way is needed.

Answer 2

what was the original question?

Answer 3

\[\sum_{n = 1}^{\infty}\frac{ 2n+1}{ n^2(n+1)^2 }\]

Answer 4

have you tried partial fractions?

Answer 5

\[\frac{2n+1}{n^2(n+1)^2}=\frac{1}{n^2}-\frac{1}{(n+1)^2}\]
\[\sum_{n=1}^\infty \frac{1}{n^2(n+1)^2}=(1/1-1/4)+(1/4-1/9)+(1/9-1/16)+\cdots\]

Answer 6

I've seen that split before, but I'm not sure on the algebra to reach it. My question for that would be, where did the 2n go and why is it subtraction? But I understand the sequence better as that generic telescopic form. I am unsure how to reach it though.

Answer 7

if you combine the two terms on the right hand side, you should get the expression on the left hand side.