Question

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)

Answer 1

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.

Answer 2

Thank you! do you know about the sum?

Answer 3

sum of series is -(x-4)(x-2)^2 / (x-3)^2