Alternating Series

Alternating Series

An alternating series has terms that switch sign:

$sum_{k=1}^{infty} (-1)^{k+1} a_k = a_1 - a_2 + a_3 - a_4 + cdots$

Alternating Series Test (Leibniz)

If $a_k > 0$, $a_k$ is decreasing, and $a_k o 0$, then the series converges.

Famous Example: Alternating Harmonic

1 - rac{1}{2} + rac{1}{3} - rac{1}{4} + cdots = ln(2) approx 0.6931

The partial sums oscillate around $ln(2)$, narrowing with each term.

Error Bound

The error of stopping at term $n$ is at most $|a_{n+1}|$ — the next term's absolute value. This makes alternating series easy to approximate with controlled accuracy.

Comparison

The partial sums zigzag toward $ln(2)$, each one closer than the last.

Your Task

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