Use the series for f(x)=1/(1-x) to write the series for g(x)=Ln|1-x|
a) C+ Ln|1+ x + x^2 + x^3+......|
b) C+ 2+ 2x+ 3x^2
c) C+ x+ x^2/2 +x^3/3
d) None of these
Note: you said since ln(1-x)= integral dx/(1-x) and integrate the taylor series for 1/(1-x). How can I do this???? This is my last exercise to submit my assigment. Please help me pleaseeeee
1/(1-x) = 1+x+x^2+x^3+x^4+...
so,
ln(1-x) = ∫(1+x+x^2+x^3+x^4+...) dx
= x + x^2/2 + x^3/3 + x^4/4 + ...
oops. Forgot the minus sign
∫1/(1-x) dx = -ln(1-x)
so the series is all negative terms.
But I'm sure you caught that ...
so will be the answer C??????
To find the series representation for g(x) = ln|1-x|, we can start with the series representation for f(x) = 1/(1-x). Recall that the series representation for f(x) is given by:
f(x) = 1 + x + x^2 + x^3 + ...
Now, we want to use this series to find the series representation for g(x) = ln|1-x|. We know that ln(1-x) is the integral (anti-derivative) of 1/(1-x) with respect to x. So, to find the series representation for g(x), we need to integrate the series representation for f(x).
To do this, let's integrate term by term. Starting with the series for f(x):
f(x) = 1 + x + x^2 + x^3 + ...
Integrating each term term by term:
∫f(x) dx = ∫(1 + x + x^2 + x^3 + ...) dx
Using the power rule for integration:
∫f(x) dx = x + (1/2)x^2 + (1/3)x^3 + (1/4)x^4 + ...
This is the series representation for the integral of f(x). However, we want to find the series representation for ln|1-x|. Notice that the integral above is equal to:
ln(1-x) + C
where C is the constant of integration. Since we're looking for the series representation of g(x) = ln|1-x|, we need to find the value of C.
To find C, we can evaluate the series representation for g(x) at some specific value of x. Let's choose x = 0. Plugging x = 0 into the series representation:
g(0) = ln|1-0| + C
g(0) = ln(1) + C
g(0) = 0 + C
g(0) = C
So, the constant of integration C is equal to g(0).
Now, we can write the series representation for g(x):
g(x) = ln|1-x| + C
g(x) = ln|1-x| + g(0)
Therefore, the correct answer is:
a) C+ Ln|1+ x + x^2 + x^3+......|