anonymous
  • anonymous
Power series problem. (x-2)^n/(n^n) Find the radius and interval of convergence.
Calculus1
  • Stacey Warren - Expert brainly.com
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schrodinger
  • schrodinger
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aykayyy
  • aykayyy
i actually just learned this today lol..... try the ratio test!
anonymous
  • anonymous
I did, but am not sure if I am doing ti correctly
anonymous
  • anonymous
I would do the nth root test?

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aykayyy
  • aykayyy
\[\lim_{n \rightarrow \infty} \left| \frac{ a _{n+1} }{ a _{n} } \right|\]
anonymous
  • anonymous
|dw:1375316023379:dw|
aykayyy
  • aykayyy
oh yes @sarahusher is probably right! i didnt see the ^n.... the root test would work better
anonymous
  • anonymous
ok ill try that
anonymous
  • anonymous
ok i did the root test and got (x-2)/n
aykayyy
  • aykayyy
then take the limit of that
anonymous
  • anonymous
Yep!
anonymous
  • anonymous
i pull the x-2 out, and the limit of (1/n) would be 0
anonymous
  • anonymous
exactly!
anonymous
  • anonymous
but then i multiply the 0 by the (x-2) right?
anonymous
  • anonymous
which would give me 0 overall?
aykayyy
  • aykayyy
yup
anonymous
  • anonymous
As limit < 1, the series will converge for every 'x'
anonymous
  • anonymous
Im not sure how they are getting the interval of convergence which they are saying is from negative infinity to positive infinity
anonymous
  • anonymous
Okay, I'll explain: So when finding the radius of convergence: We know that as the limit=0<1, the series is convergent for every 'x' So for any x for any 'x' you get (ie from -infinity to +infinity) the series will converge Using the limit that you have, you get to ROC = infinity which directly gives you the interval as -infinity
anonymous
  • anonymous
Does that make sense?
anonymous
  • anonymous
Everything else makes sense, but the ROC (x-2) Lim as 'n'-------> infinity of (1/n) = 0 0*(x-2)= 0 Im just not sure how the ROC is infiinty
anonymous
  • anonymous
'The radius of convergence' r is a 'nonnegative real number' or '∞' such that the series converges if [x+L] < r , here L is your limit So our limit is 0 so we want an 'r' such that all of our 'x'
anonymous
  • anonymous
|dw:1375317501046:dw| Think of it like a disc, no matter where your value of 'X' lies in the disc, for this example the series will always converge
anonymous
  • anonymous
ok that makes more sense now...thanks sara :)
anonymous
  • anonymous
Okay, If anything isn't clear let me know :)

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