Determine the radius of convergence, the interval of convergence, and
the sum of the series
Summation from k=2 to ∞ of
The Attempt at a Solution
possibly take the derrivitive of the power series, then find the sum then integrate?
use the ratio test to determine interval of convergence
(k+1) |x-2|^(k+2) / k|x-2|^(k+1) = [(k+1)/k] |x-2| ---> |x-2| as k ---> infinity.
Therefore the series converges for |x - 2|< 1 so 1 < x < 3 .
So the radius of convergence is 1
Is this linked to the sum? For example, if i think of taking the derivative of this function, you get f'(x)=sum from k=2 to infinity, of k(k+1)(x-2)^k which looks like the geometric series…But im not sure how to combine the ks? when 0<r<1, the geometric series converges to 1/1-r.