Series expansion

Printable View

January 13th 2009, 02:01 PM
Haris

Series expansion

Write down the series expansion of ln(1 +x^2/100), and state the radius of convergence.

Could someone explain how to solve this? Cheers x
January 13th 2009, 03:09 PM
Danny

Quote:

Originally Posted by Haris

Write down the series expansion of ln(1 +x^2/100), and state the radius of convergence.

Could someone explain how to solve this? Cheers x

Start with the expansion for $\ln ( 1 + x)$ and substitute for x

$x\; \rightarrow\; \frac{x^2}{100}$ .

For the radius of convergence, start with the radius of convergence for $\ln ( 1 + x)$ and make the same replacement (this will give another inequality to be solved).
January 17th 2009, 08:58 AM
Haris

$ln(1+x)=x-\frac{1}{2}(x)^2+\frac{1}{3}(x)^3-...$

when $x=\frac{x^2}{100}$

$ln(1+\frac{x^2}{100})=\frac{x^2}{100}-\frac{1}{2}(\frac{x^2}{100})^2+\frac{1}{3}(\frac{x ^2}{100})^3-...$

I think that's right. I still don't understand how to calculate the radius of convergence, R?
January 17th 2009, 10:49 AM
Haris

Can anyone explain what the radius of convergence is and how to calculate it given a series expansion?
January 17th 2009, 10:58 AM
o_O

Do you know the series expansion for: $f(y) = \ln (1 + y) = \sum_{i =1}^{\infty} (-1)^{n+1} \frac{y^n}{n}$ with a radius of convergence of: $-1 < y \leq 1$

Here let: $y = \frac{x^2}{100}$ and conclusion follows.
January 23rd 2009, 07:58 AM
xkcd

Radius of convergence

Hey, I'm stuck on the radius of convergence for this problem too, can someone help? I followed the first part, the expansion fine but I'm a bit lost at the end.

Many thanks...

Nevermind, I think I've got the answer (modulus of x<10?)

I have a different question now though: Expand f(x)=e^(sinx) as far as the term in x^4, and state the radius of convergence
Can I simply use the expansion of e^x and sub in sinx for each x? And do the same for the radius of convergence?

Any help much appreciated.
January 23rd 2009, 09:39 AM
Danny

Quote:

Originally Posted by xkcd

Hey, I'm stuck on the radius of convergence for this problem too, can someone help? I followed the first part, the expansion fine but I'm a bit lost at the end.

Many thanks...

Nevermind, I think I've got the answer (modulus of x<10?)

I have a different question now though: Expand f(x)=e^(sinx) as far as the term in x^4, and state the radius of convergence
Can I simply use the expansion of e^x and sub in sinx for each x? And do the same for the radius of convergence?

Any help much appreciated.

Post as a new post.
January 23rd 2009, 10:08 AM
xkcd

Expand f(x)=e^(sinx) as far as the term in x^4, and state the radius of convergence.

Can I simply use the expansion of e^x and sub in sinx for each x?
This has given me:
1 + sinx + ((sinx)^2)/2 + ((sinx)^3)/6 + ((sinx)^4)/24

This does not strike me as being correct as I do not have an x^4 term, only a (sinx)^4

And do the same for the radius of convergence?
These functions lack radii of convergence, which is also causing me to doubt my method.

Apologies for only editing, my knowledge of internet forum etiquette is limited.
If you meant that I should start a new thread as it is a new question, I will and sorry for the inconvenience.

May I also get conformation on the radius of convergence for the question posed by the opening poster?