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
limk→∞(k+1)!(2k)!k!(2(k+1))!=limk→∞(k+1)(2k)!(2k+2)!=limk→∞k+1(2k+2)(2k+1)=limk→∞12(2k+1)=0.
Hence, the series converges by the ratio test.
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