Calculus II please answer all the questions with detail explanation

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llltkl | College Teacher | (Level 3) Valedictorian

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1. The given series is:

`sum_(n=1)^ooln(n/(n+1))`

The terms in the given series can be rewritten as:

`ln(n/(n+1))= lnn-ln(n+1)`

a) Therefore, the nth partial sum,

`S_n=a_1+a_2+a_3+ . . . +a_n`

`=(ln1-ln2)+(ln2-ln3)+(ln3-ln4)+ . . . +(lnn-ln(n+1))`

`=ln1-ln2+ln2-ln3+ln3-ln4+ . . . +lnn-ln(n+1)`

`=ln1-ln(n+1)`

`=-ln(n+1)`

b) Sum of the given series is:

`sum_(n=1)^ooln(n/(n+1))`

`=lim_(n->oo)S_n`

`=lim_(n->oo) (-ln(n+1))`

`=-oo`

Hence the given series diverges.

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