Adding Series, What is wrong here?

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In summary, the student attempted to solve an equation by transforming the series and then substituting into the equation, but made a mistake.
  • #1
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Homework Statement


With binomial expansions,
[itex]\frac{x}{1-x}[/itex] = [itex]\sum[/itex][itex]^{n=\infty}_{n=1}[/itex]xn
[itex]\frac{x}{x-1}[/itex] = [itex]\sum[/itex][itex]^{n=\infty}_{n=0}[/itex]x-n

Adding these series yields:

[itex]\sum[/itex][itex]^{n=\infty}_{n=-\infty}[/itex]xn=0
This is nonsense, but what went wrong here?

The Attempt at a Solution


Obviously, [itex]\frac{x}{1-x}[/itex]+[itex]\frac{x}{x-1}[/itex]=0 (1)

It's clear that they tried to transform [itex]\frac{x}{x-1}[/itex] = [itex]\sum[/itex][itex]^{n=\infty}_{n=0}[/itex]x-n into [itex]\frac{x}{x-1}[/itex] = [itex]\sum[/itex][itex]^{n=0}_{n=-\infty}[/itex]xn and then substitute into equation (1) and get an answer of zero.

From there, I'm not sure how in the world they manipulated that series to get a neg. infinity to show up in the limits, and where the mistake in that is.
 
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  • #2
jtleafs33 said:

Homework Statement


With binomial expansions,
[itex]\frac{x}{1-x}[/itex] = [itex]\sum[/itex][itex]^{n=\infty}_{n=1}[/itex]xn
[itex]\frac{x}{x-1}[/itex] = [itex]\sum[/itex][itex]^{n=\infty}_{n=0}[/itex]x-n
The second is wrong. x- 1= -(1- x) so the second series is just
[itex]\sum_{n=1}^\infty -x^n[/itex], not [itex]\sum_{n=1}^\infty x^{-n}[/itex].

Adding these series yields:

[itex]\sum[/itex][itex]^{n=\infty}_{n=-\infty}[/itex]xn=0
This is nonsense, but what went wrong here?

The Attempt at a Solution


Obviously, [itex]\frac{x}{1-x}[/itex]+[itex]\frac{x}{x-1}[/itex]=0 (1)

It's clear that they tried to transform [itex]\frac{x}{x-1}[/itex] = [itex]\sum[/itex][itex]^{n=\infty}_{n=0}[/itex]x-n into [itex]\frac{x}{x-1}[/itex] = [itex]\sum[/itex][itex]^{n=0}_{n=-\infty}[/itex]xn and then substitute into equation (1) and get an answer of zero.

From there, I'm not sure how in the world they manipulated that series to get a neg. infinity to show up in the limits, and where the mistake in that is.
 
  • #3
jtleafs33 said:

Homework Statement


With binomial expansions,
[itex]\frac{x}{1-x}[/itex] = [itex]\sum[/itex][itex]^{n=\infty}_{n=1}[/itex]xn
[itex]\frac{x}{x-1}[/itex] = [itex]\sum[/itex][itex]^{n=\infty}_{n=0}[/itex]x-n

Adding these series yields:

[itex]\sum[/itex][itex]^{n=\infty}_{n=-\infty}[/itex]xn=0
This is nonsense, but what went wrong here?
The first series converges only if |x| < 1. The second converges only if |x| > 1. Therefore there is no x for which you can add the series and get a meaningful result.
 
  • #4
HallsofIvy said:
The second is wrong. x- 1= -(1- x) so the second series is just
[itex]\sum_{n=1}^\infty -x^n[/itex], not [itex]\sum_{n=1}^\infty x^{-n}[/itex].

I don't understand this. I realize this myself, and if I were deriving the series myself, I'd use that technique and get the same result you. Also, when I use Maple to get the series, the result also matches yours. But, reference tables I have on hand show series matching what was printed in the homework.
 
  • #5
If |x| > 1, then
[tex]\sum_{n = 0}^{\infty} x^{-n} = \frac{1}{1 - x^{-1}} = \frac{x}{x - 1}[/tex]
so the series expansion is correct. The problem is that the region of convergence is incompatible with that of the first series.
 
  • #6
jtleafs33 said:
I don't understand this. I realize this myself, and if I were deriving the series myself, I'd use that technique and get the same result you. Also, when I use Maple to get the series, the result also matches yours. But, reference tables I have on hand show series matching what was printed in the homework.

If you expand x/(1-x) as a series in t = 1/x you will get the printed result. One expansion applies for small |x| and the other applies for large |x|.

RGV
 

1. What is a series in the context of adding?

A series refers to a sequence of numbers or terms that are added together to find a sum. It is commonly used in mathematics and physics to solve problems involving addition.

2. How do I know if I have added a series correctly?

You can check if you have added a series correctly by using a calculator or by hand to find the sum of the series. If the sum matches the expected answer, then you have added the series correctly.

3. What is the correct way to add a series?

The correct way to add a series is to start from the first term and add each subsequent term until you reach the end of the series. It is important to pay attention to the signs (+ or -) and to keep track of the decimal places if the terms are not whole numbers.

4. Why am I getting a different answer when I add a series using different methods?

This could be due to rounding errors or different methods of simplifying the series. It is important to double check your calculations and make sure you are using the same method consistently.

5. What are some common mistakes when adding a series?

Some common mistakes when adding a series include forgetting to include a term, using the wrong sign (+ or -), and not keeping track of decimal places. It is also important to make sure you are using the correct formula for the specific type of series you are adding.

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