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

$\dfrac{2n-1}{(2n-1)!} = \dfrac{2n-1}{(2n-1)(2n-2)!} = \dfrac1{(2n-2)!}$

By the ratio test, the infinite series converges, since

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