proof of ratio test
Assume . By definition such that
i.e. eventually the series becomes less than a convergent geometric series, therefore a shifted subsequence of converges by the comparison test. Note that a general sequence converges iff a shifted subsequence of converges. Therefore, by the absolute convergence theorem, the series converges.
Similarly for a shifted subsequence of becomes greater than a geometric series tending to , and so also tends to . Therefore diverges.
|Title||proof of ratio test|
|Date of creation||2013-03-22 12:24:46|
|Last modified on||2013-03-22 12:24:46|
|Last modified by||vitriol (148)|