Evaluating $\int_{-e}^e(x+x^3+x^5+\cdots+x^{999}) dx$ 
Integrate
  $$\int_{-e}^e(x+x^3+x^5+\cdots+x^{999}) dx$$

I converted the integral into geometric sum, but i cannot proceed from there
The geometric sum is 
$$\frac{x(x^{1001}-1)}{x^2-1}$$

Your original functions is and "odd function" (https://en.wikipedia.org/wiki/Even_and_odd_functions#Odd_functions) such that $f(-x) = -f(x)$. When odd functions are integrated over intervals symmetric over the origin, like $[-1, 1]$ or in your case $[-e,e]$, they cancel out and integrate to zero.
Simple example: $\int_{-a}^{a} x \ dx = 0$, and you can generalize it to $\int_{-a}^{a} x^{odd} \ dx = 0$. Summing them of course still leaves you with zero.
