Uniforms needed to exceed the log of two

Let $U_1,U_2,\dots$ be i.i.d. uniform on $(0,1)$, and let $N$ be the first index $n$ at which $U_1+\dots+U_n$ exceeds $\ln 2$. Find $E[N]$.

$2$. For a threshold $x\in(0,1)$, $P(N>n)=P(U_1+\dots+U_n\le x)=\tfrac{x^n}{n!}$, so $E[N]=\sum_{n\ge0}\tfrac{x^n}{n!}=e^{x}$. At $x=\ln2$ this is $2$.