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Tuesday, November 12, 2013

Simple Divisibility Tests

In this post we will justify some common divisibility tests that most school children are familiar with. Recall that a number such as $111$ is divisible by $3$ if and only if the sum of the digits is divisible by $3$. Since $1+1+1=3$ is divisible by $3$, we see that $111$ is divisible by three.

More generally, an integer $N$ is divisible by $3$ if and only if the sum of the digits appearing in $N$ is divisible by $3$.

Saturday, November 2, 2013

A Simple Proof that the Harmonic Series Diverges

One usually encounters the harmonic series \[ \sum_{k=1}^{\infty} \frac{1}{k} \] as an example of a series that diverges for non-obvious reasons. With $H_n$ defined to be the partial sum $H_n:=\sum_{k=1}^n \frac{1}{k}$, a typical way to prove divergence is to observe that \begin{align*} H_{2n} &= H_n + \frac{1}{n+1} + \dots + \frac{1}{2n} \\ & \geq H_n + \underbrace{\frac{1}{2n} + \dots + \frac{1}{2n}}_{n \text{ terms}} \\ & = H_n + \frac{1}{2}. \end{align*} In particular, $H_{2^n} \geq 1 + \frac{n}{2}$, so the sequence of partial sums of the harmonic series has an unbounded subsequence, whence the series diverges.