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Let $f_p(x) := (x+1)^p - x^p$.

For $p=1$ the function $f_p$ is constant. For $p>1$ it is strictly increasing and for $p<1$ it is converging to zero as $x \to \infty$.

This follows for $p \in \mathbb{N}$ from the binomial formula, but I wonder wether there is a simple argument taking into account the convexity/concavity of $x \mapsto x^p$ ?

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    A really simple calculus argument does the trick. We see $$f'_p(x) = p((x+1)^{p-1} - x^{p-1}), \,\,\,\, x \in \mathbb R$$ which is clearly positive when $p > 1$ and negative when $p < 1$. – 2017-01-09

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