# Homework Help: Use geometric series to write power series representation

1. Mar 29, 2016

### NihalRi

1. The problem statement, all variables and given/known data
Complete the proof that ln (1+x) equals its Maclaurin series for -1< x ≤ 1 in the following steps.

Use the geometric series to write down the powe series representation for 1/ (1+x) , |x| < 1

This is the part (b) of the question where in part (a)I proved that ln (1+x) equals its Maclaurin series for 0< x ≤ 1by showing the limit of the errom is zero (hense converges).
2. Relevant equations

3. The attempt at a solution
The solution is actually given, I just couldn't understand it. Shown below

1. 1/(1-x) =∑ x^k for |x|<1
2. 1/(1+x) =∑ (-x)^k
3. = ∑(-1)^k x^k
4. = 1 - x + x^2 - x^3 ........for |-x| < 1 which is for |x|<1

To be clear, I only dont understand step one, the rest is just rearranging. Specifically I dont see how the expression on the left equals the summation on the right.

Note - all the summations are to infinity starting with k= 0

2. Mar 29, 2016

### blue_leaf77

Do you know the formula for a convergent, infinite geometric series starting from unity as the first term and with the ratio $x$?

Last edited: Mar 29, 2016
3. Mar 29, 2016

### Ray Vickson

By DEFINITION,
$$\sum_{k=0}^{\infty} x^k = \lim_{n \to \infty} \sum_{k=0}^n x^k$$
if that limit exists.

You can apply high-school formulas for the finite sum, and so examine in detail what happens when $n \to \infty$. You will see immediately why having $|x| < 1$ is important. Of course, the summation $\sum (-1)^k x^k$ is obtained from that of $\sum y^k$ just by putting $y = -x$ (in case that was what was bothering you).

4. Mar 29, 2016

### NihalRi

Ahh now that you mention it...... totally get it now. Thank you