Sum of infinite series

  • #1
160
3

Homework Statement


Find the sum of the series
gif.gif


Homework Equations




The Attempt at a Solution


Not sure exactly where to start. If I move 3 outside the sum I'm left with 3*sigma(1/n*4^n), which I can rewrite to 3*sigma((1/n)*(1/4)^n), which party looks like a geometric series..Any tips?
 

Attachments

  • gif.gif
    gif.gif
    726 bytes · Views: 541

Answers and Replies

  • #2
RPinPA
Science Advisor
Homework Helper
587
328
No, it's not a geometric series. The characteristic of a geometric series is that the ratio between successive terms is a constant. The ratio ##a_{n+1}/a_n## in this case is ##(n 4^n) / [(n+1)4^{n+1}]## or ##n/[4(n+1)]##, definitely not a constant.

That said, I'm not sure off-hand what trick might work here. Here's a PDF that shows some of the tricks for transforming infinite series into something that you can work with.
http://web.math.ucsb.edu/~cmart07/Evaluating Series.pdf

I think the basic hope here would be if you can use the Taylor series approach: Can you transform this into some known Taylor series, evaluated at a particular value? Then it would be that function at that value. Or perhaps the derivative of a known Taylor series? That's where I'd start looking, but I have no specific advice to give you.

Edit: I think I found one. Look at a table of common Taylor series and find one where the n-th term has an n in the denominator. You can make that work for you. That is, the n-th term of the expansion of ##f(x)## can be the same as the n-th term of this series, for a particular x.
 
  • #3
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
10,706
1,722

Homework Statement


Find the sum of the series
View attachment 234061

Homework Equations




The Attempt at a Solution


Not sure exactly where to start. If I move 3 outside the sum I'm left with 3*sigma(1/n*4^n), which I can rewrite to 3*sigma((1/n)*(1/4)^n), which party looks like a geometric series..Any tips?

You need to figure out what is
$$F(x) = \sum_{n=1}^\infty \frac{x^n}{n}.$$ This is actually a well-known series that appears in all kinds of calculus textbooks and which you might have see already in your studies. Even if you do not recognize it, there are some standard tricks that people use when facing such series, such as looking at the series for ##dF(x)/dx## or ##\int F(x) \, dx##.
 
  • #4
Delta2
Homework Helper
Insights Author
Gold Member
4,542
1,840
When you use integration you can use definite integration (instead of indefinite) and integrate from 0 to t which essentially will make the constant of integration zero.
In the left hand side you ll have

$$\int_0^t \sum (\frac{x^n}{n})'dx=\sum \frac{t^n}{n} -\sum \frac{0^n}{n}$$

and in the right hand side

$$\int_0^t\frac{1}{1-x}dx=-ln(1-t)+ln(1-0)$$
 
  • #5
330
188
I use $$log(x)=\sum_{n=1}^{\infty}\frac{1}{n}\left (\frac {x-1}{x}\right )^n$$ for ## x>\frac {1}{2}## and set
##\frac {(x-1)}{x}=\frac {1}{4}## so ##x=\frac {4}{3}##
 

Related Threads on Sum of infinite series

  • Last Post
Replies
13
Views
1K
  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
2
Views
885
  • Last Post
Replies
4
Views
955
  • Last Post
Replies
6
Views
816
  • Last Post
Replies
5
Views
1K
  • Last Post
Replies
1
Views
985
  • Last Post
Replies
1
Views
3K
  • Last Post
Replies
6
Views
2K
  • Last Post
Replies
2
Views
1K
Top