User:Qalle2/sandbox

$$ \begin{align} & {1 \over n^1} + \ldots + {1 \over n^m} \\ = & {n ^ {m-1} \over n^m} + \ldots + {1 \over n^m} \\ = & {n ^ {m-1} + \ldots + n^0 \over n^m} \\ = & {1-n^m \over (1-n)n^m} \mathsf{\ \ \ (geometric\ series)} \\ = & {1 \over (1-n)n^m} - {n^m \over (1-n)n^m} \\ = & {1 \over (1-n)n^m} - {1 \over 1-n} \\ = & {1 \over (1-n)n^m} + {1 \over n-1} \\ \to & {1 \over n-1} \mathsf{\ \ \ when\ } m \to \infty \end{align} $$