anonymous
  • anonymous
What is the radius and interval of convergence of the following series: (3x-4)^(n) --------- n=1 and is going to infinity (4n)^(n)
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
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anonymous
  • anonymous
\[\sum_{n=1}^{\infty} (3x-4)^n / (4n)^n\]
anonymous
  • anonymous
Usually the main method is to perform the ratio test on the series, and when you have the limit in terms of x, and use the geometric series boundaries -- that the absolute value of the ratio must be less than or equal to one -- and apply it to that limit (which is the ratio).
yuki
  • yuki
@:Canyoudothis... I like he way how you drew the series lol

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anonymous
  • anonymous
...is my help a bit vague? xD
anonymous
  • anonymous
thanks yuki
anonymous
  • anonymous
@quantummodulus yeah its kinda vague
yuki
  • yuki
first you want to perform the ratio test
anonymous
  • anonymous
Okay, first perform the ratio test and get the final value of the limit in terms of x.
anonymous
  • anonymous
help me
anonymous
  • anonymous
I dont know the answer to the ratio test :(
anonymous
  • anonymous
Do you know how to perform it?
yuki
  • yuki
the nth term is \[(3x-4)^n/(4n)^n\]
anonymous
  • anonymous
is it a(n+1)/ a(n)
anonymous
  • anonymous
Okay, now apply that to each n in your original series, dividing by the original series itself.
anonymous
  • anonymous
...it might take a while (it's not a very clean process, usually) but once you do it should be in terms of x.
anonymous
  • anonymous
so would i get (4n)^(n) / (3x-4)^(n) ?
anonymous
  • anonymous
oh no, hold on
anonymous
  • anonymous
(3x-4)^(n+1) (4n)^(n) ------------ * ---------- ? (4n+1)^(n+1) (3x-4)^(n)
yuki
  • yuki
exactly
anonymous
  • anonymous
soooooo? um now what haha
yuki
  • yuki
(3x-4)^n+1 can be represented as (3x-4)*(3x-4)^n
anonymous
  • anonymous
so the one on the top and the other one would cancel?
yuki
  • yuki
yep
anonymous
  • anonymous
(3x-4)(4n)^(n) ------------ (4n+1)^(n+1)
yuki
  • yuki
now that can be written in the form\[\lim_{n \rightarrow \infty}(3x-4) * (4n)^2/(4n+1)^2\]
anonymous
  • anonymous
okay thanks for the help :) ttyl
yuki
  • yuki
since you will get |3x-4| < 1, all you have to do is to solve for the inequality. however, the endpoints are always something you have to check by hand, so you need to plug in the two endpoints to the original series to see if it converges or not, ok? let me know if you need more help.

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