The task:
There are $N+1$ urns. In the $i$th urn there are $i$ white balls and $N-i$ red balls for $i=0,\ldots,N$. We choose a random urn and then we choose $n$ times a random ball from this urn with replacement. Assume that we drew a red ball each time. What is the probability that if we draw a ball from this urn one more time it will also be red?
My solution attempt:
First restate the task and say there are $i$ red balls and $N-i$ white balls, it doesn't matter but it will simplify calculations.
The probability to draw a red ball from the $i$th urn $n$ times is: $\operatorname{P}(R_{1,\ldots,n}|U_i)=\left(\frac{i}{N}\right)^n$
The probability to choose $i$th urn for any $i$ is: $\operatorname{P}(U_i)=\frac1{N+1}$
The probability to draw a red ball from a random urn $n$ times is: $\operatorname{P}(R_{1,\ldots,n})=\sum_{i=0}^N\operatorname{P}(U_i)\operatorname{P}(R_{1,\ldots,n}|U_i)=\sum_{i=0}^N\frac1{N+1}\left(\frac iN\right)^n=\frac{\sum_{i=0}^{N}i^n}{(N+1)N^n}$
The probability that we chose the $i$th urn provided we drew a red ball $n$ times is: $\operatorname{P}(U_i|R_{1,\ldots,n})=\frac{\operatorname{P}(U_i)\operatorname{P}(R_{1,\ldots,n}|U_i)}{\operatorname{P}(R_{1,\ldots,n})}=\frac{\frac1{N+1}\left(\frac iN\right)^n}{\frac{\sum_{j=0}^{N}j^n}{(N+1)N^n}}=\frac{i^n}{\sum_{j=0}^Nj^n}$
The fact that we've already drawn $n$ red balls from an urn doesn't affect the probability that we draw one more red ball, so $\operatorname{P}((R_{n+1}|R_{1,\ldots,n})|(U_i|R_{1,\ldots,n}))=\operatorname{P}(R_{n+1}|U_i)=\frac iN$
The probability that the $n+1$th ball will also be red is: $\operatorname{P}(R_{n+1}|R_{1,\ldots,n})=\sum_{i=0}^N\operatorname{P}(U_i|R_{1,\ldots,n})\operatorname{P}((R_{n+1}|R_{1,\ldots,n})|(U_i|R_{1,\ldots,n}))=\sum_{i=0}^N\frac{i^n}{\sum_{j=0}^Nj^n}\frac iN=\frac{\sum^N_{i=0}i^{n+1}}{N\sum_{i=0}^Ni^n}$
Here is where I'm stuck.
According to answers for my other questions here ((1), (2)) this is not easily solvable. Granted I've forgotten much from my former basic courses on mathematical analysis and discrete mathematics - nevertheless, these were basic courses - I'm pretty sure I haven't had any Riemann zeta functions. However, I did have integrals and generating functions and I don't remember much from these topics.
Which is why I don't believe that a solution involving Riemann zeta functions is the expected one. However, I find it possible that the expected solution involves some kind of a crude approximation, akin to the ones that were already present at this course.
How to move on with this task? Or, where is my error in the above solution attempt, if there is any?
