Partial Sums of Series

Partial Sums of Series

An infinite series $sum a_n$ is defined as the limit of partial sums:

$S_n = a_1 + a_2 + cdots + a_n = sum_{k=1}^n a_k$

If $lim_{n o infty} S_n = L$, the series converges to $L$.

Famous Series

Harmonic series (diverges):

sum rac{1}{k} = 1 + rac{1}{2} + rac{1}{3} + rac{1}{4} + cdots o infty

Basel problem (converges to $pi^2/6 approx 1.6449$):

sum rac{1}{k^2} = 1 + rac{1}{4} + rac{1}{9} + rac{1}{16} + cdots

Telescoping (exact finite sum):

sum_{k=1}^n rac{1}{k(k+1)} = 1 - rac{1}{n+1}

Because rac{1}{k(k+1)} = rac{1}{k} - rac{1}{k+1}, most terms cancel.

Convergence Test: Divergence

If $lim_{n o infty} a_n eq 0$, the series must diverge. (But $a_n o 0$ alone doesn't guarantee convergence — the harmonic series is the classic counterexample.)

Your Task

Implement double partial_sum(double (*a_n)(int), int n) that computes $sum_{k=1}^n a_n(k)$.