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the Series converges, and its sum

Evaluate `sum_(n=1)^(oo)nsin(1/n) `


This series diverges.


We use the limit test: if the series converges the limit of the nth term must be 0 as n grows without bound.


`lim_(n->oo)nsin(1/n)=1 `


We use L'Hopital's rule as we have the indeterminant form `oo * 0 `



`lim_(n->oo)n*sin(1/n)= `


`lim_(n->oo) sin(1/n)/(1/n) = 0/0`


Use L'Hopital's rule to get:


`lim_(n->oo) = (-cos(1/n)/(1/n^2))/(-1/n^2) = cos(1/n)`


`lim_(n->oo)cos(1/n)=1`

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