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QUESTION

Determine whether or not the following sequence is convergent\\
$\ds a_n=\frac{1+n(-1)^n}{n}$ for $n=1,2,\ldots$

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ANSWER

$\ds a_n=\frac{1+n(-1)^n}{n}=\frac{1}{n}+(-1)^n.$  As $n\to\infty$
the $\ds \frac{1}{n}$ term tends to 0 but\\
$(-1)^n=\left\{\begin{array}{cc}+1&n\textrm{ even}\\-1&n\textrm{
odd}\end{array}\right.$\\ Thus the terms of the sequence alternate
between +1 and $-1$, approximately, so the sequence is not
convergent.




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