Taylor series etc.

[Melissa00]

Hello again

I've spent some time looking over this problem, but I can't put any sense into this. The calculations are somewhat clear, but why do I have to go through those steps and what do the results mean

I would seriously appreciate if someone shed light on this for me


Task: Show that |q|<1 is valid with the expression above.

I have detailed instructions on the steps to solve this:
1) Find taylor series,...
which starts out with this equations
with X₀=0

and we end up with this result



2) Use the cauchy product (which leads to this result)



3) Lastly I'm supposed to integrate the equation and then take the derivative



which leads to this result

 

[pka]

Hello again

I've spent some time looking over this problem, but I can't put any sense into this. The calculations are somewhat clear, but why do I have to go through those steps and what do the results mean
I would seriously appreciate if someone shed light on this for me
View attachment 3058
Task: Show that |q|<1 is valid with the expression above.

If \(\displaystyle |q|<1\)

\(\displaystyle \sum\limits_{n = 0}^\infty {{q^n}} = \dfrac{1}{{1 - q}}\)

\(\displaystyle \sum\limits_{n = 0}^\infty {{q^{n+1}}} = \dfrac{q}{{1 - q}}\text{ multiply by }q.\)

\(\displaystyle \sum\limits_{n = 0}^\infty {{(n+1)q^n}} = \dfrac{1}{{(1 - q)^2}}\text{ first derivative}\)

 

[HallsofIvy]

Hello again

I've spent some time looking over this problem, but I can't put any sense into this. The calculations are somewhat clear, but why do I have to go through those steps and what do the results mean

I would seriously appreciate if someone shed light on this for me
View attachment 3058

Task: Show that |q|<1 is valid with the expression above.

That's a strange statement. Was it not "show that the expression above is valid with |q|< 1"?

 

[Melissa00]

Hi pka,

thanks I can't believe it's that simple, considering all the complicated steps from the answer key I received from the TA :-D

 

[Melissa00]

That's a strange statement. Was it not "show that the expression above is valid with |q|< 1"?

I know right? I didn't understand it either, didn't have a chance to go and ask someone thought...