El_Arrow
  • El_Arrow
Help! I need to find a power series representation for the function
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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El_Arrow
  • El_Arrow
f(x)=x^2/(1-7x)^2
El_Arrow
  • El_Arrow
@Michele_Laino @Concentrationalizing @jim_thompson5910
El_Arrow
  • El_Arrow
this is what i have so far |dw:1437181408215:dw|

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El_Arrow
  • El_Arrow
dont know if its right or not
anonymous
  • anonymous
That's not correct. You cannot use the geometric series on that since you still have the denominator squared. You would have to eliminate that squared denominator before you go to that.
El_Arrow
  • El_Arrow
how do i do that?
anonymous
  • anonymous
You can ignore the \(x^{2}\) and integrate \(\frac{1}{(1-7x)^{2}}\)
El_Arrow
  • El_Arrow
so have |dw:1437182143406:dw|
El_Arrow
  • El_Arrow
what do i do after that?
anonymous
  • anonymous
Thats what you got for the power series representation of the antiderivative?
El_Arrow
  • El_Arrow
oh no
anonymous
  • anonymous
Yeah, basically what I'm having you do is: \[\int\limits_{}^{}\frac{ 1 }{ (1-7x)^{2} }dx = \sum_{n=0}^{\infty}??\]
El_Arrow
  • El_Arrow
i got 1/7 ln(1-7x)
anonymous
  • anonymous
Shouldn't have ln in your answer.
El_Arrow
  • El_Arrow
|dw:1437182482351:dw|
anonymous
  • anonymous
Right, there ya go. Now of course you need + C, but that won't be an issue with what we are doing. So what is the power series representation of that function?
El_Arrow
  • El_Arrow
would it be |dw:1437182579940:dw|
anonymous
  • anonymous
Yep. I like to separate everything with power series, though, makes it easier to see simplification a lot of the time. So I would have it written like: \[\sum_{n=0}^{\infty}7^{n-1}x^{n}\] Okay, so this series is equal to the antiderivative of what we started with. SO we want to go backwards and get back to where we started, which means I need to differentiate my result. You've seen how to differentiate and integrate series?
El_Arrow
  • El_Arrow
no
El_Arrow
  • El_Arrow
so this is the power series representation |dw:1437182937373:dw|
El_Arrow
  • El_Arrow
so we need to find the radius now
anonymous
  • anonymous
Well, you essentially take the derivative as normal, you worry about the variable terms while constants just stay as is. So kind of informally, the derivative would be something like this: \[\sum_{n=0}^{\infty}7^{n-1}\frac{ d }{ dx }(x^{n})\] Where n is a constant. So what's rhe derivative of \(x^{n}\)?
El_Arrow
  • El_Arrow
nx^n
El_Arrow
  • El_Arrow
right?
anonymous
  • anonymous
\(nx^{n-1}\)
El_Arrow
  • El_Arrow
why minus 1?
El_Arrow
  • El_Arrow
oh nvm
anonymous
  • anonymous
But one thing we must also do is raise the index from 0 to 1. This is because, as is, letting the first n be 0 would make the series 0, so we fix that by raising the index by one. Thus taking the derivative would give us: \[\sum_{n=1}^{\infty}7^{n-1}nx^{n-1}\] Now just multiply in the \(x^{2}\)
El_Arrow
  • El_Arrow
like this |dw:1437183623856:dw|
El_Arrow
  • El_Arrow
cause the x^-1 and x^2 add each other right?
anonymous
  • anonymous
Yep. Just have to make the index 1 on your series.
El_Arrow
  • El_Arrow
so using the power series representation how do i determine its radius of convergence?
El_Arrow
  • El_Arrow
@dumbcow is the radius 1/7?
dumbcow
  • dumbcow
yes the radius of convergence is 1/7

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