anonymous
  • anonymous
Determine the radius of convergence, the interval of convergence, and the sum of the series: summation from k=2 to infinite of k(x-2)^(k+1)
Mathematics
schrodinger
  • schrodinger
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anonymous
  • anonymous
Apply the Ratio test. a(n+1) /a(n) = (k+1)(x-2)^(k+2) / k (x-2)^(k+1) = (k+1)/k (x-2) The series converges if | (k+1) / k (x-2) | < 1 lim k-->infinity (k+1)/k = 1 |x-2| < 1 Therefore the radius of convergence is 1 Therefore, the series converges if | x-2 | < 1 That is : -1 < x-2 < 1 add 2 throughout: 1 < x < 3 (1,3) is the interval of convergence.
anonymous
  • anonymous
Thank you! do you know about the sum?
anonymous
  • anonymous
sum of series is -(x-4)(x-2)^2 / (x-3)^2

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