= K-theory Euler class of the tangent bundle of complex projective space
{c}
{title2=$e^K(T\mathbb{CP}^n)=(n+1)t^n$}
Let $\gamma$ be the tautological line bundle and put $t=1-[\gamma]$. The <Euler sequence on complex projective space> gives
$$
T\mathbb{CP}^n\oplus\mathbb C\cong(n+1)\overline\gamma.
$$
Hence
$$
\lambda_s(\overline{T\mathbb{CP}^n})
=\frac{(1+s\gamma)^{n+1}}{1+s},
$$
and evaluating at $s=-1$ in $K^0(\mathbb{CP}^n)=\mathbb Z[t]/(t^{n+1})$ gives
$$
e^K(T\mathbb{CP}^n)=(n+1)[\gamma](1-[\gamma])^n=(n+1)t^n.
$$
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