Taylor and Maclaurin Series Lecture Video ↗

1. General Coefficient Problem

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

2. Operations on Known Maclaurin Series

Perform operations on a known Maclaurin series (using the standard 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)
Find series for: (i) \(xe^{-3x}\), (ii) \(4x \sin(2x)\), and (iii) \(2x \cos(x^2)\) Solution ↗

3. Using the Maclaurin Series Definition

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. Taylor Series at Specific Centers

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 ↗