Find a power series, centered @ x=0, for function f(x)=x/(1+2x).
I know this is a maclaurin series, but my work doesn't get the right answer. Can you please show steps? Also,do all power series start with a 1, as in (1+2x+4x^2+...)?
Thanks in advance!
No, take a look at the definition of the Maclaurin series. It starts with f(0). very power series starts with the first term of its defined sequence.
f = x/(1+2x)
f' = 1/(1+2x)^2
f(2) = -4/(1+2x)^3
f(3) = 24/(1+2x)^4
f(4) = -192/(1+2x)^5
from then on it's just the power rule:
f(n) = (-2)^(n-1) n! (1+2x)^-(n+1)
so, we have
f(0) + 1/1! f(1)(0) + 1/2! f(2)(0) + ...
= 0 + 1x - 4/2!x^2 + 24/3!x^3 - 192/4!x^4 + ...
= 0 + x - 2x^2 + 4x^3 - 8x^4 + ...
To find the power series representation of the function f(x) = x/(1 + 2x), centered at x = 0, we can use the concept of Maclaurin series.
Before we begin, let me clarify that not all power series start with a 1. The starting term depends on the function itself and the value of x at which the series is centered.
Now, let's work through the steps to find the power series representation of f(x):
Step 1: Determine the general form of the series.
Since f(x) is a rational function, we can use the geometric series formula to rewrite it. The geometric series formula is given by:
1/(1 - r) = 1 + r + r^2 + r^3 + ...
In our case, we can rewrite f(x) as:
f(x) = x * [1/(1 + (-2x))] = x * [1 - 2x + (2x)^2 - (2x)^3 + ...]
Step 2: Simplify the series.
Let's simplify the series we obtained in step 1:
f(x) = x * [1 - 2x + 4x^2 - 8x^3 + ...]
Step 3: Write the series in sigma notation.
To express the series compactly, we can use sigma notation. The series can be written as:
f(x) = ∑(-1)^n * (2x)^n
where the summation (∑) is taken over n from 0 to infinity.
So, the power series representation of f(x) = x/(1 + 2x), centered at x = 0, is:
f(x) = x * ∑(-1)^n * (2x)^n
I hope this clarifies the steps to find the power series representation of the given function. Let me know if you have any further questions!