Question

Taylor series for ln(2+3x^4)?
Find the coefficient on x^44 in the Taylor series for f(x) based at b = 0.

Answer 1

first, the taylor series at b=0 of the ln function
\[\ln(1+y)=\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}y^n\]

Answer 2

next is to rewrite the given function
\[\ln(2+3x^4)=\ln(2\cdot(1+\frac{3x^4}{2}))=\ln2+\ln(1+\frac{3x^4}{2})\]

Answer 3

using y=3x^4/2,
\[\ln(2+3x^4)=\ln2+\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}(\frac{3x^4}{2})^n\]

Answer 4

the 12th term of the taylor series (n=11) gives you the desired coefficient