Interchanging limits without Fubini/Tonelli? I'm working on a problem and have reached the following $$ \int_{0}^{1}\sum_{n=0}^{\infty}\frac{(x\log x)^{n}}{n!}\;dx $$ and I'd like to interchange the integral and the sum so that I have $$ \sum_{n=0}^{\infty}\int_{0}^{1}\frac{(x\log x)^{n}}{n!}\;dx $$ in order to solve the problem. After doing a bit of research, I've found the Fubini/Tonelli theorems mentioned a fair bit on the web and on MSE. There are some really good answers on this site, however: The problem I'm working on is for an introductory real analysis course. We've not touched measure theory and are discouraged from using results outside of the course that we have not proved. I'm curious to know whether for this particular case one could justify the interchanging of the integral and the sum without measure theory? Thank you for your time.

**Hint** : prove that the series $\sum_{n=0}^{\infty}\frac{(x\log x)^{n}}{n!}$ converges uniformly in $[0,1]$.