I have a question where i couldn't find any clue.
The question is
11⋅2+1⋅31⋅2⋅3⋅4+1⋅3⋅51⋅2⋅3⋅4⋅5⋅6+⋯
I could get the general term as tn=1⋅3⋅5⋅7⋯(2n−1)1⋅2⋅3⋅4⋅5⋅6⋯2n
I have also tried it to form the sequence in the telescopic form.But couldn't get. Any hint will be appreciated.
Answer
tn=1⋅3⋅5⋅7…(2n−1)1⋅2⋅3⋅4⋅5⋅6…2n=(2n−1)!!(2n−1)!!(2n)!!=1(2n)!!=12nn!
Therefore this sum does not go to infinity.
Invoking ex=∑∞n=0xnn!, we can compute the sum: ∞∑n=1(12)nn!=∞∑n=0(12)nn!−1=e12−1
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