Calculus 2: Taylor and Maclaurin series

Find the Taylor polynomial of degree n at x = C.

Using the Maclaurin series for $\cos(x)$, rewrite the series to accommodate $\cos(2x)$, and simplify the expression as necessary.

Find the Maclaurin series for the function $f(x) = e^x$.

Find the Taylor series for the function $f(x) = e^x$ centered at $a = 2$.

Use series to find the limit of $\frac{e^x - 1 - x}{x^2}$ as $x \to 0$.

Find the 100th derivative of $f(x) = x^3 e^{2x}$ at $x = 0$.

Find the 100th derivative of $f(x) = x^3 e^{2x}$ at $x = 0$.

Find the first three non-zero terms in the Maclaurin series for $e^x \sin x$.

Find the Maclaurin series that represents the function $f(x) = e^x$.

Evaluate the limit of $\displaystyle \frac{e^x - \cos x}{x^2}$ as $x \to 0$.

Find the sum of the series $\sum_{n=0}^{\infty} \frac{1}{n!}$.

Find the fourth degree Maclaurin polynomial for $f(x) = e^x$ and use it to approximate $e^2$.

Using the Taylor series, decompose the function $e^x$.

Compute the Taylor series for $e^x$ centered at $x = 0$.