Expected value $=\infty$?

Question

If we let $U_1, U_2, U_3,..., U_n$ be uniform (0,1), find $$\mathbb E[\sum_{i=0}^n iU_i^{i-1}]$$which, using the linearity of expectation, gives $$\sum_{i=0}^n \mathbb E[i U_i^{i-1}]$$

Doing this summation gives us $$\sum_{n=1}^\infty 1=1+1+1+...$$ Which is equal to $\infty$, or rather equals infinty in the limit $n\to \infty$.

How can an expected value be infinte? How would one give this answer if asked to 'find' the expected value? Does this mean that the expected value does not exist?

EDIT: I did a re-calculation; as far as I can see, this sum does equal $1+1+1+...$

Can anyone tell me where my calculation is incorrect? If $\Bbb E[U]=\frac{b-a}{2}$ when $U$ is distributed on $(a,b)$, then surely $\Bbb E[2U]=\frac{2(b-a)}{2}=b-a$, as does $\Bbb E[3U^2]$, and so on. Am I correct?

Answer 1

The sum is finite ($i$ goes from $1$ to $n$), so you should get a finite answer.

In other cases, expected value can diverge. In this case you might say the expected value doesn't exist, but you'd more likely say the expected value diverges to infinity.

Answer 2

"How can an expected value be infinte?"

Easily. There are many natural random variables with the expected value = $\infty$.

"How would one give this answer if asked to 'find' the expected value?"

It depends.

"Does this mean that the expected value does not exist?"

Some would say so. The question is, what is infinity? In some settings it is a well defined value. In some it is not.

But - You have a finite sum at the beginning. How it suddenly becomes infinite?