# Divergent

Today’s post is about Divergent and why the Harmonic Series is Divergent.

Divergent is a 2014 American science fiction action film directed by Neil Burger, based on the novel of the same name by Veronica Roth. The film is produced by Lucy Fisher, Pouya Shabazian and Douglas Wick, with a screenplay by Evan Daugherty and Vanessa Taylor.[4] It stars Shailene Woodley, Theo James, Zoë Kravitz, Ansel Elgort, Maggie Q, Mekhi Phifer, Jai Courtney, Miles Teller, and Kate Winslet.[5][6] The story takes place in a dystopian post-apocalyptic version of Chicago. Divergent was released on March 21, 2014 in the United States. (Wikipedia)

How does Divergent relate to Mathematics? In Mathematics, we say a series is divergent, when it does not converge to a certain number.

Some series are convergent, for instance the series $\displaystyle 1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\cdots$ which converges to 2.

There is a famous series that does not converge: The Harmonic Series.

The Harmonic Series is the divergent infinite series:

$\displaystyle\boxed{\sum_{n=1}^\infty \frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\cdots}$

Now, how can we prove that the Harmonic Series is Divergent?

We can compare the Harmonic Series with another series that is clearly Divergent.

$\displaystyle\boxed{1+\frac{1}{2}+(\frac{1}{3}+\frac{1}{4})+(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8})+(\frac{1}{9}+\cdots }>$

$\displaystyle\boxed{1+\frac{1}{2}+(\frac{1}{4}+\frac{1}{4})+(\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8})+(\frac{1}{16}+\cdots}$

The series at the bottom is clearly Divergent as it is 1+1/2+1/2+1/2+1/2+…

We can group the numbers as such because $\displaystyle 2+2^2+2^3+\cdots+2^n=\frac{2(2^{n}-1)}{2-1}=2^{n+1}-2$. Hence, after taking n of the bracketed groups above, we will end up at the $2^{n+1}$-th number.