Final answer:
To find the Taylor series for xln(1+4x), substitute -x with 4x in the Taylor series expansion for ln(1-x).
Step-by-step explanation:
To find the Taylor series for xln(1+4x), we can start with the Taylor series for ln(1-x). The Taylor series expansion for ln(1-x) is given by:
-x - (x^2)/2 - (x^3)/3 - ...
Now, to find the Taylor series for xln(1+4x), we can substitute -x with 4x in the above series:
-4x - (4x^2)/2 - (4x^3)/3 - ...
This is the Taylor series for xln(1+4x).