infinite series

Question

evaluate ∫(e^x)/x dx as an infinite series

Answer 1

So we need to be able to use e^x as an infinite Taylor Series Polynomial since we know how polynomial terms can be integrated...

Using e^x as a polynomial representation --> sum_{n=0}^{∞} xⁿ/n!

So we can simplify the integrand as sum_{n=0}^{∞} xⁿ/n! / x = sum_{n=0}^{∞} x^n-1/n!

So when we integrate we integrate term by term which gives us...

sum_{n=0}^{∞} xⁿ/(n * n!) = ∫(e^x/x) dx