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If $$\sum_{n=1}^{\infty}a_{n}$$divergent, determine whether the series $$\sum_{n=1}^{+\infty}\left(1+\dfrac{1}{n}\right)a_{n}$$ is convergent or divergent.

I know I have to use the ratio test.

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    $\begingroup$ Disconvergent, is not a word. Divergent? $\endgroup$ Oct 19 '15 at 2:20
  • $\begingroup$ "I know I have to use the ratio test" Who told you so? $\endgroup$
    – Did
    Oct 19 '15 at 6:47
  • $\begingroup$ This post should not be the duplicate of linked post. We can not directly apply the method in the linked post because the Cauchy sum only works for convergence only. But we can prove it by proving the converse as the my post shows. $\endgroup$
    – hermes
    Oct 19 '15 at 7:12
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Yes, it is divergent. We prove it by proving the converse, i.e. if $$ \sum_{n=1}^{+\infty}\left(1+\dfrac{1}{n}\right)a_{n} $$ is convergent, then $\sum_{n=1}^{+\infty}a_{n}$ is convergent.

Let $b_n=\frac{n}{n+1}$. Then $b_n$ is bounded and monotonic increasing. By post Too simple proof for convergence of $\sum_n a_n b_n$?, we can prove that $$ \sum_{n=1}^{+\infty}a_{n}=\sum_{n=1}^{+\infty}\left(1+\dfrac{1}{n}\right)a_{n}b_n $$ is convergent.

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