Determine whether series converges or diverges.
(1*3*5***(2n-1))/n!
I believe you use the ratio test. It is the algebra in this that I do not know how to do. I don't know what 1*3*5*** means, and I do not know how to approach or solve the problem.
Determine whether series converges or diverges.
(1*3*5***(2n-1))/n!
I believe you use the ratio test. It is the algebra in this that I do not know how to do. I don't know what 1*3*5*** means, and I do not know how to approach or solve the problem.
$\displaystyle 1*3*5***(2n-1)$ means multiplication of first $\displaystyle n$ odd numbers from $\displaystyle 1$ to $\displaystyle (2n-1)$.
In general it is shown as
$\displaystyle \prod\limits_{k=1}^{n}(2k-1)$.
I guess that your series is
$\displaystyle \sum\limits_{n=1}^{\infty}\dfrac{1\times3\times\cd ots\times(2n-1)}{n!}$,
which can be also represented as
$\displaystyle \sum\limits_{n=1}^{\infty}\dfrac{1}{n!}\prod\limit s_{k=1}^{n}(2k-1)$.
Your general term for this series is
$\displaystyle a_{n}:=\dfrac{1\times3\times\cdots\times(2n-1)}{n!}$.
So that, we have
$\displaystyle \dfrac{a_{n+1}}{a_{n}}=\dfrac{\dfrac{1\times3\time s\cdots\times\big(2(n+1)-1\big)}{(n+1)!}}{\dfrac{1\times3\times\cdots\times (2n-1)}{n!}}$
.........$\displaystyle =\dfrac{\dfrac{1\times3\times\cdots\times(2n+1)}{( n+1)!}}{\dfrac{1\times3\times\cdots\times(2n-1)}{n!}}$
.........$\displaystyle =\dfrac{\dfrac{1\times3\times\cdots\times(2n-1)\times(2n+1)}{n!(n+1)}}{\dfrac{1\times3\times\cd ots\times(2n-1)}{n!}}$
.........$\displaystyle =\dfrac{2n+1}{n}$
By using the ration test, we get
$\displaystyle \lim\limits_{n\to\infty}\dfrac{a_{n+1}}{a_{n}}=\li m\limits_{n\to\infty}\dfrac{2n+1}{n}$
..................$\displaystyle =2>1$,
which shows that your series is divergent.
The general term of the series is...
$\displaystyle \displaystyle a_{n} = \frac{1}{1}\ \frac{3}{2}\ \frac{5}{3}... \frac{2n-1}{n}$ (1)
... and because is $\displaystyle \displaystyle \lim_{n \rightarrow \infty} a_{n} = \infty$ the series [strongly] diverges...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$