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# convergence or divergence

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Discuss convergence or divergence of the series whose nth term is
〖(-1)〗^n n^n/((〖n+1)〗^(n+1) ) (b) 〖(-1)〗^n 〖(n+1)〗^n/n^n
(c) n^n/((〖n+1)〗^(n+1) ) (d) 〖(n+1)〗^n/n^(n+1)

Given that ∑▒a_n is a convergent series of real numbers, Prove either ∑▒b_n . Is convergent or give a counter example, when we define b_n by
a_n/n

n^(1/n)/a_n

a_n sin⁡n

a_n/((1+|a_n |))

√(a_n )/n where a_n≥0

√(a_n/n) where a_n≥0

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#### Solution Preview

Discuss convergence or divergence of the series whose nth term is
〖(-1)〗^n n^n/((〖n+1)〗^(n+1) ) (b) 〖(-1)〗^n 〖(n+1)〗^n/n^n
(c) n^n/((〖n+1)〗^(n+1) ) (d) 〖(n+1)〗^n/n^(n+1)
(a) Let . Hence and .
We consider as for all natural n. Hence the sequence is monotone decreasing sequence. And . Hence by Leibnitz's test, is convergent.
(b) Let . Hence and ...

#### Solution Summary

Discuss convergence or divergence of the series.

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