Question

Use the ratio test to determine whether the series is convergent or divergent. 1+3/1*2*3+5/1*2*3*4*5+…

Answer 1

The
`n`-th term
`(n\ge1)` in the series is evidently

`(2n-1)/((2n-1)!) = (2n-1)/((2n-1)(2n-2)!) = \frac1{(2n-2)!}`

By the ratio test, the infinite series converges, since

`\displaystyle \lim_(n\to\infty) \left| \frac{\frac1{(2(n+1)-2)!}}{\frac1{(2n-2)!}}\right| = \lim_(n\to\infty) ((2n-2)!)/((2n)!) = \lim_(n\to\infty) \frac1{2n(2n-1)} = 0 < 1`