I am having trouble figuring out the expected value in situations were the examples are going to infinity.
I have a fair dice with 8 sides. I keep a counter ($k$) of the rounds I play. Each round I increase the counter by 1 and I roll the die. I keep doing this until I roll a 1 or an 8.
So I'll define a random variable $X$ to be the amount of rounds played. I know that the odds of rolling a 1 or a 8 on a 8 sided die is $\frac1 4$ otherwise it is $\frac3 4$. The way I am trying to solve the expected value is by using a geometric series. This is where I am getting stuck I think it should look like this:
$E(X)=1*\frac1 4+2*\frac1 4+3*\frac1 4+...+n*\frac1 4$
I am unsure how to turn this into a sum.