convergence of sequence

Posting a detailed solution....

Let U_n = (n+1n-1)ⁿ be the nth term of the series.
As the nth term involves the power of n, we use what is called "Cauchy's root test" to determine convergence or divergence of series.

As per the test, Lim of n→∞ (U_n)^1/n must be greater than 1 for the series to be convergent.
However Lim(n→∞) (U_n)^1/n = 1.
Hence Cauchy's root test fails to determine the nature of series.

Trying another method,
U_n = (1 + 1/n1 - 1/n)ⁿ
= (1 + 1/n)ⁿ[(1 + (-1)/n)^n/(-1)]^-1

Evaluating the limit of U_n when n→∞,
Lim = e/e^-1 = e² = non-zero value.

Hence by Cauchy's fundamental test for divergence, Lim(n→∞) U_n ≠0, so the series is divergent.

Correct me if I'm wrong somewhere...

@pritish: he is asking to prove the convergence of the sequence and NOT the convergence of the SERIES.

Convergence of a series is different from the convergence of a sequence.