# Infinite Summations

Looking for ways to solve infinite summations, I found an ancient topic here talking about solving infinite summations that come out to answers with pi.
How would I solve an infinite summation that does not come out to an answer with pi?

Such as:
$$\sum_{n=1}^{\infty}\frac{n+1}{6^n}$$

The solution is 11/25, btw.
My attempt: I am not really experienced with this area of math, so what I did was put it into my TI-nSpire but it couldn't do it because it's not the CAS version. I plugged in a large number such as 999 terms instead of infinity terms and it came out to the right answer, but I am looking for a more "correct" way to solve the problem. I also thought of finding the sum of a geometric sequence but I realized that doesn't really work for most summations.

Oh, I'm also new to these forums, so hi to everybody! =D

EDIT: The n under the summation should say n=1

Last edited:

morphism
Homework Helper
Many of these problems can be solved by recognizing that your series is actually a special case of a more general one.

One of the more useful series is the geometric series: If |x|<1,
$$\sum_{n=0}^{\infty} x^n = \frac{1}{1-x}.$$

It's not immediately obvious how we can use this to evaluate your series. However, there's a nice trick: Differentiate both sides of the above equation term by term to get (for |x| < 1):
$$\sum_{n=1}^{\infty} n x^{n-1} = \frac{1}{(1-x)^2}.$$

Can you take it from here?

• 1 person