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So either your book is wrong or else you are misreading it.

for positive , and so2)

an=(1+n)/(2+n^2)

book says convergent but i think divergent

this time i think an>(1+n)/(n+n^2)=1/n

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Therefore .

Again, either your book is wrong or else you're misreading it.

It is not true that .3) an=1/[n ln(n)] convergent

compare with un=1/n^2 so an/un=n/ln(n) which tends to 1

Instead, use the integral test. Letting , we have

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So diverges.

Notice that for positive . So4) an=ln(n)/n^2..

i think convergent but not sure how to show this.

, and

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Therefore .

This does indeed follow from (4).5) an=ln (n)/(2n^3-1)

as for 4. think convergent but not doing correct comparison.

Use inequalities instead of trying to multiply out. For example,5) an=(2n-1)/n(n+1)(n+2)

this converges, i put into partial fractions and got an expression for the sum. Is there a nice comparison to show it quicker?

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And of course the comparison test will work quite easily here.