p-Series

A p-series has the form:

$sum_{k=1}^{infty} rac{1}{k^p} = 1 + rac{1}{2^p} + rac{1}{3^p} + cdots$

The p-series converges if and only if $p > 1$ .

$p$	Series	Converges?	Limit
$1$	harmonic $sum 1/k$	No	$infty$
$2$	Basel $sum 1/k^2$	Yes	$pi^2/6 approx 1.6449$
$3$	Apéry $sum 1/k^3$	Yes	$approx 1.2021$
$4$	$sum 1/k^4$	Yes	$pi^4/90 approx 1.0823$

Group the harmonic series:

ight) + left( rac{1}{5}+cdots+ rac{1}{8} ight) + cdots$$ Each group sums to at least $ rac{1}{2}$, so the total diverges. ### Why p=2 Converges The Basel problem — Euler showed the sum equals $ rac{pi^2}{6}$ using the factorization of $ rac{sin(x)}{x}$. Numerically, the partial sums converge slowly: $S_{1000} approx 1.6439$, approaching $1.6449$. ### Your Task Implement `double p_series(int p, int n)` that computes $sum_{k=1}^n rac{1}{k^p}$. Use a loop to compute $k^p$ without `math.h`.

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