Question

(a) Evaluate (e^x-1)/x
as an infinite series.

Answer 1

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.