# Determine whether the series $\sum_{n=1}^{+\infty}\left(1+\frac{1}{n}\right)a_{n}$ is convergent or divergent [duplicate]

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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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.