91影视

In the following exercises, compute the Taylor series of each function around \(x=1\). $$ f(x)=e^{2 x} $$

In the following exercises, find the Maclaurin series of \(F(x)=\int_{0}^{x} f(t) d t\) by integrating the Maclaurin series of \(f\) term by term. If \(f\) is not strictly defined at zero, you may substitute the value of the Maclaurin series at zero. $$ F(x)=\int_{0}^{x} \frac{\ln (1+t)}{t} d t ; f(t)=\sum_{n=0}^{\infty}(-1)^{n} \frac{t^{n}}{n+1} $$

Given center \(a\), and identify its interval of convergence. $$ f(x)=\frac{1}{1-2 x} ; a=0 $$

Given center \(a\), and identify its interval of convergence. $$ f(x)=\frac{1}{1-4 x^{2}} ; a=0 $$

In the following exercises, compute at least the first three nonzero terms (not necessarily a quadratic polynomial) of the Maclaurin series of f. $$ f(x)=\sin \left(x+\frac{\pi}{4}\right)=\sin x \cos \left(\frac{\pi}{4}\right)+\cos x \sin \left(\frac{\pi}{4}\right) $$

In the following exercises, integrate the given series expansion of \(f\) term- by-term from zero to \(x\) to obtain the corresponding series expansion for the indefinite integral of \(f\). \(f(x)=\frac{2 x}{1+x^{2}}=2 \sum_{n=0}^{\infty}(-1)^{n} x^{2 n+1}\)

Given center \(a\), and identify its interval of convergence. $$ f(x)=\frac{x^{2}}{1-4 x^{2}} ; a=0 $$

In the following exercises, evaluate each infinite series by identifying it as the value of a derivative or integral of geometric series. Evaluate \(\sum_{n=1}^{\infty} \frac{n}{2^{n}}\) as \(f^{\prime}\left(\frac{1}{2}\right)\) where \(f(x)=\sum_{n=0}^{\infty} x^{n} .\)

In the following exercises, compute at least the first three nonzero terms (not necessarily a quadratic polynomial) of the Maclaurin series of f. $$ f(x)=\tan x $$

In the following exercises, compute at least the first three nonzero terms (not necessarily a quadratic polynomial) of the Maclaurin series of f. $$ f(x)=\ln (\cos x) $$