Geometric_Series_Test

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Definition

The geometric series $\sum_{n=0}^{\infty} ar^n$ converges to $\frac{a}{1-r}$ if $|r| < 1$ and diverges if $|r| \geq 1$.

Proof

For $|r| < 1$, the partial sums are $S_n = a \cdot \frac{1 - r^{n+1}}{1 - r}$.

Since $|r| < 1$, we have $r^{n+1} \to 0$ as $n \to \infty$.

Thus $S_n \to \frac{a}{1-r}$.

For $|r| \geq 1$, the terms $ar^n$ do not converge to $0$, so the series diverges.