Choose a random number between 00 and 11 and record its value. Keep doing it until the sum of the numbers exceeds 11. How many tries do we need?

Choose a random number between $0$ and $1$ and record its value. Do this again and add the second number to the first number. Keep doing this until the sum of the numbers exceeds $1$. What’s the expected value of the number of random numbers needed to accomplish this?

Here is a (perhaps) more elementary method. Let $X$ be the amount of numbers you need to add until the sum exceeds $1$. Then (by linearity of expectation):
Now $X > k$ if the sum of the first $k$ numbers $x_1,\ldots,x_k$ is smaller than $1$. This is exactly equal to the volume of the $k$-dimensional set:
This is known as the $k$-dimensional simplex. When $k = 1$, we get a line segment of length $1$. When $k = 2$, we get a right isosceles triangle with sides of length $1$, so the area is $1/2$. When $k=3$, we get a triangular pyramid (tetrahedron) with unit sides, so the volume is $1/6$. In general, the volume is $1/k!$, and so