Let $f$ be the function defined by $$f(x) = \sum_{n=1}^{+\infty} \left(\frac{1}{n+x} - \frac{1}{n}\right)$$ Let $k \in \mathbb{N}^{*}$. For all $x \in \mathcal{D}_{f}$, verify that $x + k \in \mathcal{D}_{f}$, then calculate $f(x+k) - f(x)$.
Let $f$ be the function defined by
$$f(x) = \sum_{n=1}^{+\infty} \left(\frac{1}{n+x} - \frac{1}{n}\right)$$
Let $k \in \mathbb{N}^{*}$. For all $x \in \mathcal{D}_{f}$, verify that $x + k \in \mathcal{D}_{f}$, then calculate $f(x+k) - f(x)$.