In the question they want me to use the Ratio test to find out whether the series (2n)factorial/n^(2n) converges or not. The n = 1 and goes to infinity. everytime I try solving it I keep getting a infinity > 1 which means it (diverges), but the answer in the textbook says 4/e^2 < 1 and it converges. Someone please help me understand where the 4/e^2 came from?
Advanced Calculus (Ratio test)
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LCKurtz
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In the ratio test, the part from the [MATH]n^{2n}[/MATH] will give
[MATH]\frac{n^{2n}}{(n+1)^{2(n+1)}} = \frac{n^{2n}}{(n+1)^{2n}(n+1)^2}=\left (\frac{n}{(n+1)}\right)^{2n}\cdot \frac{1}{(n+1)^2}= \left( \frac{1}{1+\frac 1 n} \right )^{2n}\cdot \frac{1}{(n+1)^2}[/MATH]You might recognize that first factor as having limit [MATH]\frac 1 {e^2}[/MATH]. There is more, but maybe you can take it from there.
[MATH]\frac{n^{2n}}{(n+1)^{2(n+1)}} = \frac{n^{2n}}{(n+1)^{2n}(n+1)^2}=\left (\frac{n}{(n+1)}\right)^{2n}\cdot \frac{1}{(n+1)^2}= \left( \frac{1}{1+\frac 1 n} \right )^{2n}\cdot \frac{1}{(n+1)^2}[/MATH]You might recognize that first factor as having limit [MATH]\frac 1 {e^2}[/MATH]. There is more, but maybe you can take it from there.