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Calculate the limit of the queue an, using increasing criterion....

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Calculate the limit of the queue an, using increasing criterion.

an=sin(1!)/(n^2+1)+sin(2!)/(n^2+2)+...+sin(n!)/(n^2+n)

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For using the increasing criterion, we have to determine another string bn, with the general term

bn=1/(n^2+1)+1/(n^2+2)+......+1/(n^2+n)

So, n/(n^2+n)<bn<n/(n^2+1)

lim bn= lim n/(n^2+n)=lim n/(n^2+1)=0, when n is extending to infinity.

But module (an)<bn, and lim bn=0, so lim an=0

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