Power series Construction Help

  • #1
90
3
Very basic issue here.

Using:

[tex] \frac{1}{1-x} = \sum_{i=0}^{\infty} x^{i} , |x|<0 [/tex]

Find the power series representation and interval of convergence for:

[tex]f(x) = \frac{1}{(1-3x)^{2}} [/tex]

We have that:

[tex] \frac{d}{dx}\left[\frac{1}{1-x}\right] = \frac{1}{(1-x)^{2}} = \sum_{i=0}^{\infty} ix^{i-1} , |x| < 0 [/tex]

Of all things, the algebra of manipulating the function in question is stumping me. For some reason whatever I try doesn't seem to work.
 

Answers and Replies

  • #2
34,149
5,764
Very basic issue here.

Using:

[tex] \frac{1}{1-x} = \sum_{i=0}^{\infty} x^{i} , |x|<0 [/tex]

Find the power series representation and interval of convergence for:

[tex]f(x) = \frac{1}{(1-3x)^{2}} [/tex]

We have that:

[tex] \frac{d}{dx}\left[\frac{1}{1-x}\right] = \frac{1}{(1-x)^{2}} = \sum_{i=0}^{\infty} ix^{i-1} , |x| < 0 [/tex]
Do you understand what they're doing here? Particularly in getting the summation at the end? This is basically the same as what you need to do with your series.

Also your inequality at the end is wrong. ##|x| \ge 0## for all real x.
Morgan Chafe said:
Of all things, the algebra of manipulating the function in question is stumping me. For some reason whatever I try doesn't seem to work.
 
  • #3
Math_QED
Science Advisor
Homework Helper
2019 Award
1,701
720
If I am correct the series converges for all x: |x|<1. When you find the derivative of this series, it can be that the series converges in either -1 and or 1, so you might want to check that.
 
  • #4
Ssnow
Gold Member
523
154
yes, the initial series converges for ##|x|<1##. Observe that in the second case ##x## will be replaced by ##3x##, you must consider that ...
 

Related Threads on Power series Construction Help

Replies
1
Views
1K
Replies
1
Views
524
Replies
1
Views
2K
  • Last Post
Replies
4
Views
704
Replies
6
Views
1K
Replies
1
Views
2K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
3
Views
6K
  • Last Post
Replies
13
Views
2K
  • Last Post
Replies
13
Views
3K
Top