Determine if the series
Infinite summation (n=1 to inf) of (-1)^n-1 * (1*3*5...(2n-1)) / (2n-1)!
is absolutely convergent, conditionally convergent or divergent.
Your answers are very much appreciated!
Anonymous
Favorite Answer
Use the Ratio test.
We see that:
a_n = [(-1)^(n - 1) * 1 * 3 * 5 * ... * (2n - 1)]/(2n - 1)!
a_(n + 1) = [(-1)^n * 1 * 3 * 5 * ... * (2n - 1) * (2n + 1)]/(2n + 1)!.
By the ratio test:
lim (n-->infinity) a_(n + 1)/a_n
==> lim (n-->infinity) {[(-1)^n * 1 * 3 * 5 * ... * (2n - 1) * (2n + 1)]/(2n + 1)!} / {[(-1)^(n - 1) * 1 * 3 * 5 * ... * (2n - 1)]/(2n - 1)!}
= lim (n-->infinity) {[(-1)^n * (2n + 1)]/(2n + 1)!}/[(-1)^(n - 1)/(2n - 1)!]
= -1 * lim (n-->infinity) [(2n + 1)/(2n + 1)!]/(2n - 1)!
= -1 * lim (n-->infinity) (2n + 1)/[(2n - 1)! * (2n + 1)!]
= 0.
Therefore, the series converges absolutely.
I hope this helps!