Use geometric series to write power series representation

[NihalRi]

Homework Statement

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).

Homework Equations

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 don't understand step one, the rest is just rearranging. Specifically I don't see how the expression on the left equals the summation on the right.

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

 

[blue_leaf77]

Specifically I don't see how the expression on the left equals the summation on the right.

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

 

[Ray Vickson]

Homework Statement

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).

Homework Equations

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 don't understand step one, the rest is just rearranging. Specifically I don't see how the expression on the left equals the summation on the right.

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

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).

 

[NihalRi]

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

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