Real series/Ratio test/Fact/Proof

Proof

The convergence does not change (though the sum) when we change finitely many members of the series. Therefore, we can assume ${}k_{0}=0$. Moreover, we can assume that all ${}a_{k}$ are positive real numbers. Then

${}a_{k}={\frac {a_{k}}{a_{k-1}}}\cdot {\frac {a_{k-1}}{a_{k-2}}}\cdots {\frac {a_{1}}{a_{0}}}\cdot a_{0}\leq a_{0}\cdot q^{k}\,.$

Hence, the convergence follows from the comparison test and the convergence of the geometric series.

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