# Is the series $\sum_{n=1}^\infty \frac{1}{1+n/2}$ convergent?

Is $$\sum_{n=1}^{\infty} \frac1 {1+\frac{n}{2}}$$ convergent? I tried using the comparison test, but all I get is that it is inferior to the harmonic series, which is divergent.

• hint: $\displaystyle \sum_{n=1}^{\infty} \frac{1}{1+\frac{n}{2}}=2\sum_{n=1}^{\infty}\frac{1}{n+2}=2\sum_{n=3}^{\infty} \frac{1}{n}$. – Galc127 Mar 31 '16 at 5:03
• You're right, thank you. – poliliol Mar 31 '16 at 5:05
• The comparison test works like a charm as well... $$\frac1{1+\frac{n}2}\geqslant\frac23\frac1n$$ – Did Mar 31 '16 at 5:06

$$\sum_{n=1}^N\frac{1}{1+n/2}=2\sum_{n=1}^N\frac{1}{n+2}=2\sum_{n=1}^{N+2}\frac1n-3$$
As an alternative if Dr. MV's approach of making it a case of the harmonic series doesn't do it for you for some reason, you can use the integral test on this series as the function $$f(x) = \frac{1}{1 + \frac{x}{2}}$$ is positive, continuous, and decreasing on the interval of $[0,\infty)$
Thus, as $$\int_{0}^{\infty} \frac{2}{2 + x} dx = 2ln(2+x) \Big|_0^\infty$$ very much so diverges, the series must also diverge.