`sum_(n=1)^oo n/(2n+3)` Verify that the infinite series diverges

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`sum_(n=1)^oo n/(2n+3)`

To verify if the series diverges, apply the nth-Term Test for Divergence. 

It states that if the sequence `a_n` does not converge to zero, then the series diverges.

`lim_(n->oo) a_n != 0 `        `:.`  `sum` `a_n`   diverges

 

Applying this, the limit of the term of the series as n approaches infinity is:

`lim_(n->oo) n/(2n + 3) `

`=lim_(n->oo) n/(n(2+3/n)) `

`= lim_(n->oo) 1/(2+3/n)`

`=(lim_(n->oo)1)/(lim_(n->oo) (2+3/n))`

`= 1/(2+0)`

`=1/2`

The limit of the series is not zero. Therefore, by the  nth-Term Test for Divergence, the series diverges.

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