Thread: Power Series Expansion

Problem
===========================
Let (*)

Evaluate this formula for using the power series expansion for f at q = 0:

(1)

===============================
I have two questions.

1.) Is this power series expansion correct? I tried inputting this function in MATLAB and it gives me a different power series expansion:

(2)

2. Regarding the power series which is defined as

at q = a.

Since this power series (1) I do not understand how a power series can be derived this way since at q = 0, then f(0) from (*) is undefined because we can not divide by 0, correct?

Thank you for reading.

Originally Posted by Paperwings

Problem
===========================
Let (*)

Evaluate this formula for using the power series expansion for f at q = 0:

(1)

===============================
I have two questions.

1.) Is this power series expansion correct? I tried inputting this function in MATLAB and it gives me a different power series expansion:

(2)

2. Regarding the power series which is defined as

at q = a.

Since this power series (1) I do not understand how a power series can be derived this way since at q = 0, then f(0) from (*) is undefined because we can not divide by 0, correct?

Thank you for reading.

The Matlab expansion is correct. Your problem statement has an error in the term. Notice that the other terms agree.

The singularity of f at 0 is "removable", so there is no problem with continuity at that point. (You might try evaluting the limit as q -> 0 by l'Hopitals rule, although it's obvious from the power series.) It's a little like the function ; it only "looks like" there is a problem at 0.