Partial sums of powers

Function computes exact or approximate solutions to partial sums of real non-negative powers via power series.

The sequence S_k(n) = 1^k + 2^k + 3^k ... + n^k is computed via power series by the function call

Skn = partial_sum(k,n,R)

with input constraints:
real, non-negative k
real n>=1
integer R between 1-20 specifies approximation depth

Output is necessarily an approximation for non-integer k. Output is exact for integer k, assuming R>=k+1.

Example: approximate 1^(pi/4) + 2^(pi/4) + 3^(pi/4) ... + 27^(pi/4) using a depth of 3.
Skn = Skn(pi/4,27,3)
Answer: Skn = 207.8504...