Find a power series for the function

[CarlaNemz]

Hello everyone I've watched tons of videos to try and figure this one out but I keep getting it wrong, the second photo is of a question which i believe relates to it because it looks practically the same. Im lost.

 

[blamocur]

Have you studied Taylor series (https://en.wikipedia.org/wiki/Taylor_series) ?

 

[Dr.Peterson]

Hello everyone I've watched tons of videos to try and figure this one out but I keep getting it wrong, the second photo is of a question which i believe relates to it because it looks practically the same. Im lost.

View attachment 30081View attachment 30082

I think this lesson should help:

Calculus II - Power Series and Functions

In this section we discuss how the formula for a convergent Geometric Series can be used to represent some functions as power series. To use the Geometric Series formula, the function must be able to be put into a specific form, which is often impossible. However, use of this formula does...

tutorial.math.lamar.edu

Yours, like the first couple examples there, can be derived from the geometric series. It will require a slightly different start; so please make some attempt and show us your work, so we can nudge you in the right direction. (We much prefer nudging to telling you what to do.)

In fact, since you say you "keep getting it wrong", I can tell that you already have some work to show us. Did you forget to read this?

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[Angelcat]

Have you tried: $$\sum_{n=0}^{\infty}(-x-k)^n\newline$$when k is a positive integer?