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Question

Show that the infinite series
u1+u2+u3+u4+......
is convergent or divergent according as limnnun is <1 or >1.

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Solution

Let un be the nth term of the series.
Let for all nN (N= a natural number)
n|un|k1
|un|kn1
The geometric series n=Nkn converges at k<12
n=N|un| also converges by comparison test
Hence
If \underset{n\rightarrow \infty}{\lim}\sqrt[n]{| u_n|} \lt 1$, the series converges
And If n|un|>1 for n tends to infinity, then un fails to converge, hence the series is divergent.

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