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!
answered in link below
To find the power series representation for the function f(x) = x/(1+2x) centered at x = 0, we can use the geometric series expansion.
Step 1: Find the formula for the geometric series
The geometric series is given by:
1 / (1 - r) = 1 + r + r^2 + r^3 + ...
Step 2: Convert the function to a form that matches the geometric series
First, let's rewrite f(x) as:
f(x) = x * (1 / (1 + 2x))
We can rewrite 1 / (1 + 2x) as:
1 / (1 - (-2x))
So now f(x) can be rewritten as:
f(x) = x * (1 - (-2x))^(-1)
Step 3: Apply the geometric series formula
Now, we can apply the geometric series formula to (1 - (-2x))^(-1):
(1 - (-2x))^(-1) = 1 + (-2x) + (-2x)^2 + (-2x)^3 + ...
Step 4: Multiply by x
Multiply each term by x to obtain the power series for f(x):
f(x) = x + (-2x^2) + (-2x)^3 + ...
Step 5: Simplify the terms
(-2x^2) can be written as -2x^2
(-2x)^3 can be simplified as -8x^3
So the power series representation, centered at x = 0, for f(x) = x/(1 + 2x) is:
f(x) = x - 2x^2 - 8x^3 + ...
Regarding the second part of your question, not all power series start with 1. The terms in a power series can have any coefficients, depending on the specific function being represented. In this case, the power series for f(x) does not start with 1, but with x.