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