Final answer:
The infinite series representation of (e^x - 1)/x is 1 + x/2! + x^2/3! + x^3/4! + ... after applying Taylor series expansion for e^x and simplifying each term by x.
Step-by-step explanation:
To evaluate (e^x - 1)/x as an infinite series, we can start by considering the Taylor series expansion of e^x, which is:
e^x = 1 + x + x^2/2! + x^3/3! + ...
Subtracting 1 from both sides and then dividing by x, we get:
(e^x - 1)/x = (1 + x + x^2/2! + x^3/3! + ... - 1)/x
=(x + x^2/2! + x^3/3! + ...)/x
Now, dividing each term by x, we have:
(e^x - 1)/x = 1 + x/2! + x^2/3! + x^3/4! + ...
This is the infinite series representation of the original expression.