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.
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5$\begingroup$ hint: $\displaystyle \sum_{n=1}^{\infty} \frac{1}{1+\frac{n}{2}}=2\sum_{n=1}^{\infty}\frac{1}{n+2}=2\sum_{n=3}^{\infty} \frac{1}{n}$. $\endgroup$ – Galc127 Mar 31 '16 at 5:03
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$\begingroup$ You're right, thank you. $\endgroup$ – poliliol Mar 31 '16 at 5:05
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2$\begingroup$ The comparison test works like a charm as well... $$\frac1{1+\frac{n}2}\geqslant\frac23\frac1n$$ $\endgroup$ – Did Mar 31 '16 at 5:06
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$$\sum_{n=1}^N\frac{1}{1+n/2}=2\sum_{n=1}^N\frac{1}{n+2}=2\sum_{n=1}^{N+2}\frac1n-3$$
Therefore, inasmuch as the harmonic series diverges, the series of interest does also.
answered Mar 31 '16 at 5:06
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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.
