Geometric Series

A geometric series has each term a constant multiple of the previous:

$sum_{k=0}^{n-1} a cdot r^k = a + ar + ar^2 + cdots + ar^{n-1}$

For $r eq 1$ :

$S_n = a cdot rac{1 - r^n}{1 - r}$

The infinite geometric series $sum a cdot r^k$ converges if and only if $|r| < 1$ :

$sum_{k=0}^{infty} a cdot r^k = rac{a}{1 - r} quad ext{when } |r| < 1$

$r$	Series behavior
$r = 0.5$	converges to $2a$
$r = 1$	diverges (grows linearly)
$r = 2$	diverges (grows exponentially)
$r = -1$	oscillates (diverges)

$a = 1$ , $r = 0.5$ , $n = 10$ :

ight) approx 1.9980$$ The infinite sum: $ rac{1}{1-0.5} = 2$ — the partial sum already reached $1.998$. ### Your Task Implement `double geometric_partial(double a, double r, int n)` that computes $sum_{k=0}^{n-1} a cdot r^k$ using a loop.

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