`sum_(n=1)^oo (1/n-1/(n+2))` Determine the convergence or divergence of the series.

`sum_(n=1)^oo(1/n-1/(n+2))`

`S_n=(1/1-1/(1+2))+(1/2-1/(2+2))+(1/3-1/(3+2))+..........+(1/(n-1)-1/(n+1))+(1/n-1/(n+2))`

`S_n=(1-1/3)+(1/2-1/4)+(1/3-1/5)+.......+(1/(n-1)-1/(n+1))+(1/n-1/(n+2))`

This sum is a telescoping series,

`S_n=(1+1/2-1/(n+1)-1/(n+2))`

`sum_(n=1)^oo(1/n-1/(n+2))=lim_(n->oo)S_n`

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

`=(1+1/2)`

`=3/2`

So the series converges.

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