Real series/Ratio test/Fact

Ratio test

Let

$\sum _{k=0}^{\infty }a_{k}$

be a series of real numbers. Suppose there exists a real number ${}q$ with ${}0\leq q<1$, and a ${}k_{0}$ with

${}\vert {\frac {a_{k+1}}{a_{k}}}\vert \leq q\,$

for all ${}k\geq k_{0}$ (in particular ${}a_{k}\neq 0$ for ${}k\geq k_{0}$).

Then the series ${}\sum _{k=0}^{\infty }a_{k}$ converges absolutely.

Proof, Write another proof