# Maclaurin Series using Substitution

1. Apr 17, 2014

### alanwhite

1. The problem statement, all variables and given/known data
Use a known Maclaurin series to compute the Maclaurin series for the function: f(x) = x/(1-4(x^2))

2. Relevant equations
1/(1-x) = ∑x^n

3. The attempt at a solution
I tried removing x from the numerator for: x ∑ 1/(1-4(x^2)), which would end up through substitution as x ∑ (4^n)(x^2n). Not too sure this is correct use of substitution however.

2. Apr 17, 2014

### szynkasz

Yes, it is correct.

3. Apr 17, 2014

### Staff: Mentor

What you did makes sense, but how you described what you did doesn't make sense. If you can find the series for 1/(1 - 4x2), just multiply term-by-term to get the series for x/(1 - 4x2). Pulling a variable out of a summation that involves x isn't a valid operation.
For example,
$$\sum_{n = 1}^k n^2 \neq n \cdot \sum_{n = 1}^k n$$

4. Apr 18, 2014

### alanwhite

So in essence, I would write the terms of the series ∑ (4^n)(x^2n) and multiply each term by x? Alright, is there no way of writing the series so that there is no x variable outside of the summation?

5. Apr 18, 2014

### SammyS

Staff Emeritus
Mark did say to take the sum, then multiply through by x, term by term. (Basically, that's the distributive law, and the extra x will be inside of the sum.

For example,

$\displaystyle x\left(\sum _{n=1}^ k x^n \right) = \sum _{n=1}^ k x^{n+1}$