A divergent series …. or maybe not?

The number $n$ ranges over all possible powers with both the base and the exponent positive integers greater than $n$ , assuming each such value only once. Prove that:

$\sum_{n} \frac{1}{n-1}=1$

Let us denote by $\mathcal{M}$ the set of positive integers greater than $1$ that are not perfect powers ( i.e are not of the form $a^p$ , where $a$ is a positive integer and $p \geq 2$ ). Since the terms of the series are positive , we can freely permute them. Thus,

$\begin{align*} \sum_{n} \frac{1}{n-1} &= \sum_{m \in \mathcal{M}} \sum_{k=2}^{\infty} \frac{1}{m^k-1} \\ &= \sum_{m \in \mathcal{M}} \sum_{k=2}^{\infty} \sum_{j=1}^{\infty} \frac{1}{m^{kj}}\\ &=\sum_{m \in \mathcal{M}} \sum_{j=1}^{\infty} \sum_{k=2}^{\infty} \frac{1}{m^{kj}} \\ &= \sum_{m \in \mathcal{M}} \sum_{j=1}^{\infty} \frac{1}{m^j \left ( m^j-1 \right )} \\ &= \sum_{n=2}^{\infty} \frac{1}{n\left ( n-1 \right )} \\ &= \sum_{n=2}^{\infty} \left ( \frac{1}{n-1} - \frac{1}{n} \right )\\ &= 1 \end{align*}$

A binomial sum

An indefinite integral

A divergent series …. or maybe not?

They might interest you:

A binomial sum

An indefinite integral

Huygen's Inequality

An arccot sum

A convergent series

Binomial coefficients as multiple sum

On an entire function

Fourier transformation

Square of a number

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