Test the convergence of the series 1/2 + 3/4 + 4/9 + 5/16 +..........+ n+1/n² +.....

 1. Test the convergence of the series

1/2 + 3/4 + 4/9 + 5/16 +..........+ n+1/n²

 +.....

Sol:

Let the given series be denoted by ∑ un.

Then. un = n+1/n²

Let the auxiliary series ∑vn be such that vn = n/n² = 1/n

Then un/vn = (n+1)/n² × n/1 = (n+1)/n = 1+(1/n)

lim n--›∞ un/vn = lim n--›∞{1+(1/n)} = 1,

Which is a finite quantity.

Hence by comparison test, the series ∑un and ∑vn are either both convergent or both divergent.

But the auxiliary series ∑vn = ∑(1/n) is a divergent series, since ∑(1/n^p) is divergent when p= 1.

Hence the given series is also divergent.

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