What are the first few coefficients for this series?

[τheory]

Homework Statement

The function $f(x)=ln(2−x)$ is represented as a power series in the form:

$\sum C_{n}x^{n},n,0,inf$

Find the first five coefficients in the power series.

Basically, this is a problem from my online homework. I did a lot of work to get to my answers, and I feel sort of tired to type up all of it so I'll just show my final result (hopefully you guys don't mind). And if need be to know, I first differentiated f(x), then wrote down the power series for it, then integrated the power series to obtain the power series for the original f(x).

Afterwards, I expanded this power series for the first five terms and wrote down the coefficients:

$(-1/2), (-1/8), (-1/24), (-1/64), (-1/160)$I then inputted these numbers into the boxes specified on the web page and I received a message saying that they were wrong. I've checked my work twice, my TI89, and even Wolfram Alpha; I get the same answers from these sources, so can someone tell me what is wrong? Also if someone does have a solution, you do not have to tell me what the actual answers are, just notify me with what I did wrong.

 

[vela]

What about the constant term?

 

[τheory]

There was no constant term; after integrating the whole power series, I obtained x^(n+1)/(n+1) times a number. If n = 0 as the first term, then you would get x^(0+1)/(0+1), which is equal to just x.

Edit: Also if you are interested to know, this was the power series I derived:

$\sum\frac{-1}{2}\frac{1}{2^n}\frac{x^{n+1}}{n+1},n,0,∞$

 

[Ray Vickson]

There was no constant term; after integrating the whole power series, I obtained x^(n+1)/(n+1) times a number. If n = 0 as the first term, then you would get x^(0+1)/(0+1), which is equal to just x.

Edit: Also if you are interested to know, this was the power series I derived:

$\sum\frac{-1}{2}\frac{1}{2^n}\frac{x^{n+1}}{n+1},n,0,∞$

There must be a constant term: what is f(0)? Is it zero? By the way, is your notation supposed to be $\sum_{n=0}^\infty C_n x^n$?

RGV

 

[τheory]

I don't think that's the case because if the derivative of the function is $f ' (x)=\frac{-1}{2-x}$, then the power series representation would be:

$\sum_{n=0}^\infty (\frac{-1}{2})(\frac{x}{2})^{n}$

If I wrote out the first five terms of this series, I would get:

$(\frac{-1}{2})-(\frac{1}{4})x-(\frac{1}{8})x^2-(\frac{1}{16})x^3-(\frac{1}{32})x^4$

Next I would have to integrate these terms to get a partial representation for $f(x) = ln(2-x)$:

$\int (\frac{-1}{2})-(\frac{1}{4})x-(\frac{1}{8})x^2-(\frac{1}{16})x^3-(\frac{1}{32})x^4 dx$
$= (\frac{-1}{2})x-(\frac{1}{8})x^2-(\frac{1}{24})x^3-(\frac{1}{64})x^4-(\frac{1}{160})x^5$

So as seen from above, the coefficients would be as I specified in my first post?

EDIT: I tried inputting 0, -1/2, -1/8, -1/24, -1/64 and I got everything except for 0 right; so now I know there must be something before -1/2? But if there is something before -1/2, how would I find it?

EDIT2: I tested ln(2) and found that to be the replacement for 0, but can someone explain to me how the first term is ln(2)?

EDIT3: Never mind, I found out why it was ln(2); the C that is resulted from integration had to have been solved, I totally neglected that so thank you guys for pointing that out.

 

[HallsofIvy]

When you integrated the series term by term, did you include the constant of integration?

 

[τheory]

Okay so here's another similar problem:

I first differentiated the function $f(x)=ln(1-x^2)$ and wrote down the power series for it:

$\sum\stackrel{∞}{n=0}(-2x)(x^2)^n$

Then I integrated the power series to get a series for the original function:

$\sum\stackrel{∞}{n=0}(-2)\frac{x^{2n+2}}{2n+2}$

There was suppose to be a constant of integration but I found it to be zero, so after expanding the series for five terms, I found the coefficients to be:

$0, -1, -1/2, -1/3, -1/4$

After inputting these numbers, as seen by the above picture, the web page said they were wrong; can someone tell me why?

 

[Dick]

Okay so here's another similar problem:
View attachment 40518

I first differentiated the function $f(x)=ln(1-x^2)$ and wrote down the power series for it:

$\sum\stackrel{∞}{n=0}(-2x)(x^2)^n$

Then I integrated the power series to get a series for the original function:

$\sum\stackrel{∞}{n=0}(-2)\frac{x^{2n+2}}{2n+2}$

There was suppose to be a constant of integration but I found it to be zero, so after expanding the series for five terms, I found the coefficients to be:

$0, -1, -1/2, -1/3, -1/4$

After inputting these numbers, as seen by the above picture, the web page said they were wrong; can someone tell me why?

I think the homework is expecting c1 to be the coefficient of x in the series. That's zero. The first nonzero term, -1, is the coefficient of x^2 in the series. That should be c2.

 

[τheory]

I tried what you said and it turns out c1 is indeed zero, meaning c0 is zero, c1 is zero, and c2 is -1. However c3 and c4 are still wrong; I used c3 = -1/2 and c4 = -1/3, and the web page said they were wrong?

 

[Dick]

I tried what you said and it turns out c1 is indeed zero, meaning c0 is zero, c1 is zero, and c2 is -1. However c3 and c4 are still wrong; I used c3 = -1/2 and c4 = -1/3, and the web page said they were wrong?

You aren't thinking about the powers of x in your series. It's -x^2-x^4/2-x^6/3-... isn't it. c3 should be the coefficient of the power of x^3. Etc.

 

[τheory]

Oh well that works, c3 is actually zero, and c4 is -1/2, but what I don't get is why these numbers don't correspond to the exponent of x. For example, x^3 corresponds to the coefficient of -1/2, but instead x^3 corresponds to the coefficient of zero?

 

[Dick]

Oh well that works, c3 is actually zero, and c4 is -1/2, but what I don't get is why these numbers don't correspond to the exponent of x. For example, x^3 corresponds to the coefficient of -1/2, but instead x^3 corresponds to the coefficient of zero?

The attachment says the series is c_n*x^n. So c1 is the coefficient of x, c2 is the coefficient of x^2, etc etc. The odd powers are missing from your series, so all of the odd coefficients should be 0. Right?

 

[τheory]

Yes that's true, but the terms that are suppose to be missing are the even terms; expanding the series produces exponents of odd numbers, that's why I'm confused as to why the even numbers are showing and the odd numbers are missing.
$\sum\stackrel{∞}{n=0}(-2)\frac{x^{2n+2}}{2n+2}$

 

[Dick]

Yes that's true, but the terms that are suppose to be missing are the even terms; expanding the series produces exponents of odd numbers, that's why I'm confused as to why the even numbers are showing and the odd numbers are missing.
$\sum\stackrel{∞}{n=0}(-2)\frac{x^{2n+2}}{2n+2}$

2n+2 is even, isn't it? That's the exponent. What do you mean 'exponents of odd numbers'?

 

[τheory]

Whoa my bad, my scratch paper was conglomerated with numerous revisions of different series, making me look at the wrong one.