Monday, 27 November 2017

calculus - Problem solving a series with convergence test: suminftyk=1frack!(2k)!



Good morning, I have a big problem solving this:
k=1k!(2k)!



I'm trying solving this limit with test of D'Alembert, but I have a problem solving the limit.



limk(k+1)!k!(2k+2)!2k!=(?)




please, help me.


Answer



We have that
\lim_{k\to\infty}\frac{(k+1)!(2k)!}{k!(2(k+1))!}=\lim_{k\to\infty}\frac{(k+1)(2k)!}{(2k+2)!}=\lim_{k\to\infty}\frac{k+1}{(2k+2)(2k+1)}=\lim_{k\to\infty}\frac1{2(2k+1)}=0.
Hence, the series converges by the ratio test.


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