# proof of ratio test

Assume $k<1$. By definition $\exists N$ such that
$n>N\implies|\frac{a_{n+1}}{a_{n}}-k|<\frac{1-k}{2}\implies|\frac{a_{n+1}}{a_{n% }}|<\frac{1+k}{2}<1$

i.e. eventually the series $|a_{n}|$ becomes less than a convergent geometric series, therefore a shifted subsequence of $|a_{n}|$ converges by the comparison test. Note that a general sequence $b_{n}$ converges iff a shifted subsequence of $b_{n}$ converges. Therefore, by the absolute convergence theorem, the series $a_{n}$ converges.

Similarly for $k>1$ a shifted subsequence of $|a_{n}|$ becomes greater than a geometric series tending to $\infty$, and so also tends to $\infty$. Therefore $a_{n}$ diverges.

Title proof of ratio test ProofOfRatioTest 2013-03-22 12:24:46 2013-03-22 12:24:46 vitriol (148) vitriol (148) 6 vitriol (148) Proof msc 40A05 msc 26A06