Convergence of Infinite Series

Question

It is possible to deduce the value of the following (in my opinion) converging infinite series? If yes, then what is it?

$$\sum_{n=1}^\infty\frac{1}{2\cdot n}$$

where n is an integer. Sorry if the notation is a bit off, I hope youse get the idea.

Answer 1

The series is not convergent, since it is half of the harmonic series which is known to be divergent$^1$.

$$\sum_{n=1}^{\infty }\frac{1}{2n}=\frac{1}{2}\sum_{n=1}^{\infty }\frac{1}{n}.$$

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$^1$ The sum of the following $k$ terms is greater or equal to $\frac{1}{2}$

$$\frac{1}{k+1}+\frac{1}{k+2}+\ldots +\frac{1}{2k-1}+\frac{1}{2k}\geq k\times \frac{1}{2k}=\frac{1}{2},$$

because each term is greater or equal to $\frac{1}{2k}$.

Answer 2

To me, the easiest way to see that the harmonic series diverges is to use the Integral test. Then you do not have to deal with coming up with a formula.