1. ## Series help!

Hey folks.

I still havent got my head round the idea of a series expansion and am stuck on the questions in my homework regard series expansion.

1) What is the series expansion of $ln(1+x^2 / 100)$ and it's radius of conversion?

There is a an x-formula in my textbook which reads $ln x = (x-1) - \frac{1}{2}(x-1)^2 + \frac{1}{3}(x-1)^3 - ...$. First of all I don't exactly understand where it has come from (the explanation isn't clear) and secondly, can I just plug in my "x" is into the formula?
I know the Radius of Convergence is when the function doesn't make sense so for the question the ROC is infinity i.e. it always makes sense?

2) Use the fact that $sin x = Im(exp(ix))$ to calculate a simpler derivation of this result.

Im not at all sure where to start...something to do with De Moivre's theorem I beleive but I've never really understood it? Could you point me in the right direction or more?

Thanks a lot people

2. Originally Posted by Solo
Hey folks.

I still havent got my head round the idea of a series expansion and am stuck on the questions in my homework regard series expansion.

1) What is the series expansion of $ln(1+x^2 / 100)$ and it's radius of conversion?

There is a an x-formula in my textbook which reads $ln x = (x-1) - \frac{1}{2}(x-1)^2 + \frac{1}{3}(x-1)^3 - ...$. First of all I don't exactly understand where it has come from (the explanation isn't clear) and secondly, can I just plug in my "x" is into the formula?
I know the Radius of Convergence is when the function doesn't make sense so for the question the ROC is infinity i.e. it always makes sense?
see here under "List of Taylor series of some common functions" and find the formula for $\ln (1 + x)$ (they write "log" instead of "ln" here). just replace $x$ with $\frac {x^2}{100}$ everywhere. it also show you how they derive it.

you must review Taylor series and MacLauren series expansions.

2) Use the fact that $sin x = Im(exp(ix))$ to calculate a simpler derivation of this result.
what result? the one in part (1)?

3. Originally Posted by Solo
2) Use the fact that $sin x = Im(exp(ix))$ to calculate a simpler derivation of this result.
How curious. The poster here posted the same problem with the same mistake: Series Expansion

4. The series for $ln(1+t)=\sum_{n=1}^{\infty}\frac{(-1)^{t+1}t^{n}}{n}$

Try that.

5. Thanks.

Ah sorry it is related to another question which I forgot to post.

(a) Find the first 4 terms in the power series expansion of $e^x sin x$ by multiplying together the series expanstions of $e^x$ and $sin x$.

(b) Use the fact that $sin x = Im(exp(ix))$ to calculate a simpler derivation of this result.

6. Originally Posted by Solo
Thanks.

Ah sorry it is related to another question which I forgot to post.

(a) Find the first 4 terms in the power series expansion of $e^x sin x$ by multiplying together the series expanstions of $e^x$ and $sin x$.

(b) Use the fact that $sin x = Im(exp(ix))$ to calculate a simpler derivation of this result.
see the link given in post #3

7. Yep. Thanks.