## gerryliyana anyone can? Find the Taylor series expansion of a function of f(z)=z exp(2z) about z = -1 and find the region of convergence., 10 months ago 10 months ago

1. gerryliyana

@ParthKohli

2. ParthKohli

I wish I had known how to do this. It wouldn't take anything more than the definition of the Taylor Series. I mightn't know how to apply it here though. :-|

3. gerryliyana

ok.., no problem parthkohli...,

4. CarlosGP

The expression of a Taylor Series of f(z) around point "a" is:$f(z)=f(a)+\sum_{n=1}^{\infty}f^{(n}(a)(z-a)/n!$ where$f^{(n}(a)$stands for the value of the n-th derivative of f(z) in z=a, which in our case is a=-1. Thus the expression for our function would be: $f(z)=f(-1)+\sum_{n=1}^{\infty}f^{(n}(-1)(z+1)^n/n!$ $f(-1)=-e^{-2}$ $f^{(1}(z)=e^{2z}(2z+1)$$f^{(2}(z)=2e^{2z}(2z+2)$$f^{(3}(z)=4e^{2z}(2z+3)$$f^{(4}(z)=8e^{2z}(2z+4)$ and so on. If you observe this four derivatives we can obtain a general expression like this:$f^{(n}(z)=2^{(n-1)}e^{2z}(2z+n)\rightarrow f^{(n}(-1)=2^{(n-1)}e^{-2}(n-2)$ If you replace this in Taylor´s expression, you get:$f(z)=-e^{-2}+e^{-2}\sum_{n=1}^{\infty}2^{(n-1)}(n-2)(z+1)^n/n!$

5. CarlosGP

In order to get the region of convergence, apply a criterium such as Alembert's and see how the series converges for any value of z, which means the region of convergence is plus/minus infinite

6. gerryliyana

@CarlosGP how about the region of convergence ??

7. niksva

in order to find the region of convergence we need to apply ration test as suggested by @CarlosGP

8. niksva

*ratio

9. gerryliyana

i thought the region convergence is $|z+1| < \infty$ hbu ??

10. niksva

11. niksva

i have tried to apply alembert ratio test look at the steps $a _{n}= 2^{n-1}\frac{ (n-2) (z+1)^{n} }{ n! }$ $a _{n+1} = 2^{n}\frac{ (n-1) (z+1)^{n+1} }{ (n+1)! }$ now $\lim_{n \rightarrow infinity} \frac{ a _{n+1} }{ a _{n} } = \lim_{n \rightarrow infinity} \frac{ 2(n-1)(z+1) }{ (n+1)(n-2) }= \frac{ 1 }{ R }$ where R is region of convergence

12. niksva

@gerryliyana is this ok?

13. gerryliyana

and R = infnty

14. niksva

yeah it will come out to be infinity after applying the limits hope u understand the region of convergence

15. gerryliyana

ah yea..., it's ok now.,