Let $n \geq 1$ be an integer, let $X = \mathbb{CP}^n$, and let $T X$ denote the holomorphic tangent bundle of $X$, viewed as a rank-$n$ complex vector bundle. Let $\mathcal O_X(1)$ be the hyperplane line bundle on $X$, and let \begin{align*} H := c_1(\mathcal O_X(1)) \in H^2_{\mathrm{dR}}(X;\mathbb C) \end{align*} be the hyperplane class, normalized by the condition that its restriction to a projective line satisfies \begin{align*} \int_{\mathbb{CP}^1} H = 1. \end{align*} Under the standard de Rham [cohomology ring](/theorems/2271) presentation \begin{align*} H^*_{\mathrm{dR}}(X;\mathbb C) \cong \mathbb C[H]/(H^{n+1}), \end{align*} the total Chern class of $T X$ is \begin{align*} c(TX) = (1+H)^{n+1} \end{align*} in $\mathbb C[H]/(H^{n+1})$. Equivalently, \begin{align*} c(TX) = \sum_{k=0}^{n} \binom{n+1}{k} H^k. \end{align*} In particular, \begin{align*} c_1(TX) = (n+1)H. \end{align*}