Taylor and Maclaurin Series   Lecture Video

1. Find the Taylor series for \(f\) centered at \(3\) if \(\displaystyle{f^{(n)}(3)=\frac{(-1)^{n} n !}{2^{n}(n+1)}}\)  solution

2. Perform operations on a known Maclaurin series (i.e., use the provided table) to find the power series representation for each function \(f\). Include the interval of convergence.

(a) \(\displaystyle f(x)= x^2 e^{-2x}\)  solution

(b) \(\displaystyle f(x)= 3\sin(2x)\)  solution

(c) \(\displaystyle f(x)= 4\cos(x^2)\)  solution

(d) (i) \(\displaystyle f(x)= xe^{-3x}\)   (ii) \(\displaystyle f(x)= 4x \sin(2x)\)   (iii) \(\displaystyle f(x)= 2x \cos(x^2) \)  solution

3. Use the Taylor series formula (the definition of a Maclaurin series) to find the Maclaurin series of the following functions.

(a) \(f(x)=4(1-x)^{-2} \)   solution

(b)  \( f(x)=\ln (1+x) \)   solution

(c)  \( f(x)=e^{3x}\)  solution

(d) \( f(x)=e^{-5x}\)  solution

(e) \(f(x)=xe^x \)   solution

4. Find  the Taylor series for \(f(x)=\cos x \) centered at \( a= \pi\).  solution

5. Find the Taylor series for \(f(x)=\frac{2}{x}\) centered at \(a=-2\).   solution

6. Find the Taylor series for \(f(x)=\ln(x)\) centered at \(a=5\).   solution