Harmonic Series

Relevant wiki: Convergence Tests

By using comparison test

$1+1/2+(1/3+1/4)+(1/5+1/6+1/7+1/8)+1/9 +\cdots > 1+1/2+(1/4+1/4)+(1/8+1/8+1/8+1/8)+(1/16+.\cdots 1/16)+ \cdots$

because each term in harmonic series is greater than or equal to to second series.

Now the second series is equal to

$1+1/2+1/2+1/2+1/2+\cdots .$

infinite. So harmonic series is also infinite and does not converge .

Take the harmonic series 1 + $\frac{1}{2}$ + $\frac{1}{3}$ + ... + $\frac{1}{x}$ , and apply the integral test on series. After integration, the resultant integral is ln(x). Then we take the limit of the result and we see lim as x -> infinity, ln(x) also approaches infinity. Since the integral does not converge, the harmonic series does not converge.

harmonic series diverges - Oresme proof

We can look at it $\frac{1}{n^p}$ as a p- series, where p is equal to 1, so it diverges.