91Ó°ÊÓ

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. $$ F(x)=\tan ^{-1} x ; \quad f(t)=\frac{1}{1+t^{2}}=\sum_{n=0}^{\infty}(-1)^{n} t^{2 n} $$

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. $$ F(x)=\tanh ^{-1} x ; \quad f(t)=\frac{1}{1-t^{2}}=\sum_{n=0}^{\infty} t^{2 n} $$

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. $$ F(x)=\int_{0}^{x} \frac{\sin t}{t} d t ; \quad f(t)=\frac{\sin t}{t}=\sum_{n=0}^{\infty}(-1)^{n} \frac{t^{2 n}}{(2 n+1) !} $$

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. $$ F(x)=\int_{0}^{x} \cos (\sqrt{t}) d t ; \quad f(t)=\sum_{n=0}^{\infty}(-1)^{n} \frac{x^{n}}{(2 n) !} $$

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. $$ F(x)=\int_{0}^{x} \frac{1-\cos t}{t^{2}} d t ; \quad f(t)=\frac{1-\cos t}{t^{2}}=\sum_{n=0}^{\infty}(-1)^{n} \frac{t^{2 n}}{(2 n+2) !} $$

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. $$ F(x)=\int_{0}^{x} \frac{\ln (1+t)}{t} d t ; \quad f(t)=\sum_{n=0}^{\infty}(-1)^{n} \frac{t^{n}}{n+1} $$

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

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

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

Compute at least the first three nonzero terms (not necessarily a quadratic polynomial) of the Maclaurin series of \(f\). $$ f(x)=e^{x} \cos x $$