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- Apr 13, 2013

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Hello!!!

I am given this exercise:

Let $a_{n}$ be a sequence of positive numbers.Show that the sequence $\sum_{n=1}^{\infty} a_{n}$ converges if and only if the sequence $\sum_{n=1}^{\infty} \frac{a_{n}}{1+a_{n}}$ converges.

That's what I have tried so far:

->We know that $\frac{a_{n}}{a_{n}+1}\leq a_{n}$ ,so from the Comparison Test,if the sequence $\sum_{n=1}^{\infty} a_{n}$ converges,then the sequence $\sum_{n=1}^{\infty} \frac{a_{n}}{1+a_{n}}$ also converges.

->If the sequence $\sum_{n=1}^{\infty} \frac{a_{n}}{1+a_{n}}$ converges, $\frac{a_{n}}{a_{n}+1} \to 0$,so there is a $n_{0}$ such that $\frac{a_{n}}{1+a_{n}}<\frac{1}{2} \forall n \geq n_{0}$ ....But how can I continue?

I am given this exercise:

Let $a_{n}$ be a sequence of positive numbers.Show that the sequence $\sum_{n=1}^{\infty} a_{n}$ converges if and only if the sequence $\sum_{n=1}^{\infty} \frac{a_{n}}{1+a_{n}}$ converges.

That's what I have tried so far:

->We know that $\frac{a_{n}}{a_{n}+1}\leq a_{n}$ ,so from the Comparison Test,if the sequence $\sum_{n=1}^{\infty} a_{n}$ converges,then the sequence $\sum_{n=1}^{\infty} \frac{a_{n}}{1+a_{n}}$ also converges.

->If the sequence $\sum_{n=1}^{\infty} \frac{a_{n}}{1+a_{n}}$ converges, $\frac{a_{n}}{a_{n}+1} \to 0$,so there is a $n_{0}$ such that $\frac{a_{n}}{1+a_{n}}<\frac{1}{2} \forall n \geq n_{0}$ ....But how can I continue?

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