Is the endomorphism $P(X)\mapsto P(X+1)-P(X)$ diagonalizable?

Let $n\in \mathbb{N}$ , $n\ge 2$ . Is the endomorphism $\varphi$ of $\mathbb{C}_n[X]$ , which maps the polynomial $P(X)$ to the polynomial $P(X+1)-P(X)$ , diagonalizable?

Show Hint:

One may write the matrix of

\varphi

with respect to a well-chosen basis or reason by contradiction and use the fundemental theorem of algebra.

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Computing

\lim\limits_{n\to+\infty}\sum_{p=n+1}^{kn} \frac1 p

for

k > 1