# The decomposition of the numerator

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1. Apr 6, 2016

### Theengr7

• Moved from a technical forum, so homework template missing
(a) Find a power series representation for the function.

I'm struggling on the decomposition of the numerator. This exercise is from chapter 8, section 6 of Th Stewart Calculus book.

2. Apr 6, 2016

### SteamKing

Staff Emeritus
It's not clear what you mean by "decomposition of the numerator".

For any fraction of the form $\frac{a+b}{D} = \frac{a}{D} + \frac{b}{D}$

3. Apr 6, 2016

### Ssnow

I don't know if is useful to decompose the numerator but rewriting the function as $f(x)=(5+x)\cdot\frac{1}{1-x}$ can help ...

4. Apr 6, 2016

### Math_QED

Use Steamking's hint, then you should recognize something rather familiar, can you write down where those series converge?

5. Apr 6, 2016

### HallsofIvy

Staff Emeritus
The very first thing I would do is actually divide the denominator into the numerator: $\frac{5+ x}{1- x}= -1+ \frac{6}{1- x}$ and then expand $\frac{6}{1- x}$ using the fact that the geometric series $\sum_{i=0}^\infty ar^n= \frac{a}{1- r}$.

6. Apr 6, 2016

### Theengr7

I do not think that this series converges; it diverges. What I got is 5+6∑(x)^n fron 0 to ∞. I do not know if I satisfy your answer.

7. Apr 6, 2016

### Math_QED

I did not check whether your answer is valid. I do know that your answer has to include a geometric series. Do you know when a geometric series converges? If not, you should look it up.

Last edited: Apr 6, 2016
8. Apr 6, 2016

### Math_QED

Use hallsofIvy's tip (easier than Steamking's one). Once you get the right answer (you can actually write it as one sum), you should also mention for what values this series converges to f(x), which is the interval where the geometric series converges.

9. Apr 6, 2016

### Theengr7

I do. My answer is right because it's an online homework, and the website accepts this answer.

10. Apr 7, 2016

### Math_QED

If so, can you send me how you got that answer? Then I can learn something new :)