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
35,287
7,140
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
member 587159
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
554
166
yes, the initial series converges for ##|x|<1##. Observe that in the second case ##x## will be replaced by ##3x##, you must consider that ...
 

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