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 <---------My answer. Can you check for me please?
d) None of these
C - x - x^2/2 - x^3/3 .....
as written I would have to say none of the above, but I bet that is an errorr in the assignment.
wat grade is this plz tell meh = What grade is this? Please tell me.
That is about year two of calculus meh, suspect AP or college student.
To find the series for g(x) using the series for f(x), you can start by considering the power series representation of the natural logarithm function, ln(1+x). This series is given by:
ln(1+x) = x - x^2/2 + x^3/3 - x^4/4 + ...
Now, let's look at the function g(x) = ln|1-x|. The absolute value sign does not change the behavior of the series, since it just removes the negative sign. Hence, we can focus on the series for ln(1+x) and substitute -x in place of x. This gives:
ln|1-x| = -x - (-x)^2/2 + (-x)^3/3 - (-x)^4/4 + ...
Simplifying the terms:
ln|1-x| = -x + x^2/2 - x^3/3 + x^4/4 - ...
Comparing this with the answer choices:
a) C + ln|1+x+x^2+x^3+...| - The function inside the logarithm is different from what we have.
b) C + 2 + 2x + 3x^2 - The terms in the series do not match.
c) C + x + x^2/2 + x^3/3 - This matches our derived series.
Therefore, the correct answer is c) C + x + x^2/2 + x^3/3.