Let $X$ be a uniformly distributed random variable between 0 and 10. Also let $Y$ be exponentially distributed with $\lambda=1$. I want to solve the following equation where $\tau\geq 0$, $$\sum_{i=1}^{\infty}\left(\int_{(i-1)\tau}^{i\tau}\int_{(i-1)\tau}^{t}tf_{X}(x)f_{Y}(t-x)dxdt+i\tau\int_{(i-1)\tau}^{\tau}\left(1-F_{Y}(i\tau-x)\right)f_{X}(x)dx\right)$$ I now use that $f_{X}(x)=\frac{1}{10}, F_{Y}(x)=1-e^{-x}$ and $f_{Y}(x)=e^{-x}$ to get the equation,
$$\sum_{i=0}^{\infty}\left(\int_{(i-1)\tau}^{i\tau}\int_{(i-1)\tau}^{t}\frac{t}{10}e^{-t+x}dx dt + i\tau\int_{(i-1)\tau}^{i\tau}e^{-i\tau+x}\cdot\frac{1}{10}dx\right).$$ I first compute the double integral to be $$\int_{(i-1)\tau}^{i\tau}\int_{(i-1)\tau}^{t}\frac{t}{10}e^{-t+x}dxdt=\frac{1}{10}\left(\frac{2i\tau^{2}-\tau^{2}}{2}-1+e^{-\tau}-i\tau+i\tau^{-\tau}+\tau\right),$$ and the other integral, $$i\tau\int_{(i-1)\tau}^{i\tau}e^{-i\tau+x}\cdot\frac{1}{10}dx=\frac{i\tau}{10}\left(1-e^{-\tau}\right).$$ This means I now need to solve the summation, $$\sum_{i=0}^{\infty}\left(\frac{1}{10}\left(\frac{2i\tau^{2}-\tau^{2}}{2}-1+e^{-\tau}-i\tau+i\tau^{-\tau}+\tau\right)+\frac{i\tau}{10}\left(1-e^{-\tau}\right)\right).$$ This can be simplified to, $$\sum_{i=0}^{\infty}\left(\frac{1}{10}\left(\frac{2i\tau^{2}-\tau^{2}}{2}-1+e^{-\tau}+\tau\right)\right).$$ This is the point where I get stuck. From looking at the sum I get the idea that this is a diverging sum, which should not be the case. Where did I make a mistake?