Proving Series Convergence: Comparing $\sum y_n$ with $\sum \frac{y_n}{1+y_n}$

[evinda]

Hello! (Wave)
We have a sequence $(y_n)$ with $y_n \geq 0$.
We assume that the series $\sum_{n=1}^{\infty} \frac{y_n}{1+y_n}$ converges. How can we show that the series $\sum_{n=1}^{\infty} y_n$ converges?

It holds that $y_n \geq \frac{y_n}{1+y_n}$.

If we would have to prove the converse we could use the comparison test. Could you give me a hint what we can do in this case?

 

[Evgeny.Makarov]

From $\dfrac{y_n}{1+y_n}\to0$ we can conclude that $y_n\to0$. Therefore, $1+y_n$ can be eventually bounded from above, say, by $3/2$. So, $\dfrac{y_n}{1+y_n}$ can be bounded from below by $\dfrac23y_n$.

 

[math771]

Hi there! Great question. One way to show that the series $\sum_{n=1}^{\infty} y_n$ converges is by using the comparison test. Since we know that $y_n \geq \frac{y_n}{1+y_n}$, we can compare the two series and see that $\sum_{n=1}^{\infty} y_n$ is bounded above by $\sum_{n=1}^{\infty} \frac{y_n}{1+y_n}$. Since the latter series converges, the former must also converge. Hope that helps!