Supposing as a nice simple example I use the power series command,

series(e^x, x=0,8)

to get,

1+x+12x² + 16x³ + 124x⁴ + 1120x⁵ + 1720x⁶ + 15040x⁷

Is there automated anyway to get this as a Sigma representation? 

browsing on math.stackexchange I came across this maple code:

asympt(int(sqrt(-k^2+1)*exp(I*k*x), k = -1 .. 1), x, 2)

that gave the correct answer:

https://math.stackexchange.com/questions/532394/how-to-analyze-the-asymptotic-behaviour-of-this-integral-function

so I wondered would it not be great if maple done asymptotic expansion of integrals in general.

say as an example Li(x).  Can maple expand Li(x) and how?