• Support PF! Buy your school textbooks, materials and every day products Here!

Trying to solve this infinite series?

  • Thread starter eNathan
  • Start date
  • #1
352
1
Trying to solve this infinite series??

Hey folks! I've spent hours trying to solve this and have exhausted all available resources.. I just need to be pointed in the right direction!

Homework Statement


Compute the sum of the infinite series (I believe this is an arithmetico geometric series):
[itex]\sum \frac{n+1}{4^{n}}[/itex]

For n=0..infinity

Homework Equations


I know the standard way to solve a geometric series, but don't know how to solve this type of series.


The Attempt at a Solution


I've set up something like this:
[itex]S_{n} = \sum \frac{n+1}{4^{n}}[/itex]
I've tried multiplying by 1/4, 4 and other logical things, but am just not sure how to proceed.

Thanks in advance!!
 

Answers and Replies

  • #2
34,281
10,322
Your expression can be written as
$$\sum_0^\infty \frac{1}{4^{n}} + \sum_1^\infty \frac{1}{4^{n}} + \sum_2^\infty \frac{1}{4^{n}} + \dots$$
(This assumes you start your sum at 0, if you start at 1 you have to modify it a bit.)
 
  • #3
haruspex
Science Advisor
Homework Helper
Insights Author
Gold Member
33,249
5,290
Another approach, perhaps more general in application, is to multiply the terms by xn then integrate. You should then be able to sum the series into closed form and differentiate to get a closed form for the original sum. Finally set x=1.
 

Related Threads on Trying to solve this infinite series?

  • Last Post
Replies
9
Views
2K
Replies
4
Views
1K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
2
Views
920
  • Last Post
Replies
4
Views
635
  • Last Post
Replies
6
Views
2K
  • Last Post
Replies
4
Views
873
  • Last Post
Replies
7
Views
2K
Top