MHB Power Series Problem: Solve f(3x) = 1/(1 - 3x)

[JProgrammer]

So here is the problem I am trying to solve:

You can combine two (or more) convergent power series on the same interval I. Using the properties of the geometric series, find the power series of the function below.

Series:
f(x) = 1/(1 - x) = sigma k = 0, infinity = 1+ x + x^2 + x^3

Function:
f(3x) = 1/(1 - 3x)

I plug in: expand sum (1/(1 - 3x) to order infinity and it returns with just understanding the 1/1 - 3x part and nothing else, producing a wrong answer.

If someone could tell me what I am doing wrong, I would appreciate it.

Thank you,

 

[Greg]

. . . it returns . . .

Hi JProgrammer, what is "it"?

I believe the correct power series would be

$$1+3x+9x^2+27x^3+81x^4+\cdots$$

convergent for $|x|<\dfrac13$.

 

[HallsofIvy]

Let u= 3x. Then f(3x)= f(u)= 1+ u+ u^2+ u^3+ \cdot\cdot\cdot= 1+ 3x+ (3x)^2+ (3x)^3+ \cdot\cdot\cdot= 1+ 3x+ 9x^2+ 27x^3+ \cdot\cdot\cdot.

"I plug in: expand sum (1/(1 - 3x) to order infinity and it returns with just understanding the 1/1 - 3x part and nothing else, producing a wrong answer."

"Plug in" to what? Don't assume that some program can replace thinking!