Exam 10: Approximating Functions Using Series

What is the general term of the series $1-\frac{x^{5}}{5 !}+\frac{x^{10}}{10 !}-\frac{x^{15}}{15 !}+\cdots$ ?

Find the first harmonic of the function $f(x)=$$\left\{\begin{array}{c}-2 \\2\end{array}\right.$ -\pi< 0< x\leq0 x\leq\pi .

Find the 12th-degree Taylor polynomial for $x \sin \left(x^{2}\right)$ centered at x = 0.Suppose you use the first two non-zero terms of the polynomial to approximate $x \sin \left(x^{2}\right)$ for 0 < x < 1.Is your approximation too big or too small?

a)Find the Taylor series for $f(x)=\frac{7}{1+x^{2}}$ using a series for $\frac{1}{1+x}$ . b)Use the series from part a)to find the Taylor series for $g(x)=7 \arctan x$ .

Suppose you approximate $\sqrt{0.9}$ and $\sqrt{0.2}$ using the Taylor polynomial of degree 3 around x = 0 for the function $f(x)=\sqrt{1-x}$ .Which approximation is more accurate?

Which gives the better approximation of $4 e^{03}$ , the Taylor polynomial about zero with three terms, or the Fourier polynomial with three terms?

It can be shown that the Maclaurin series for $e^{z}$ , $\cos z$ and $\sin z$ converge for all values of z in the complex numbers, just as they do for all values of x in the real numbers. a)Write down and simplify the Maclaurin series for $e^{i x}$ . b)Write down the Maclaurin series for $\cos x$ and $i \sin x$ c)Use the series you found in parts a)and b)to show that $e^{i x}=\cos x+i \sin x$ .(This is one of several formulas called "Euler's Formula.") d)Find the value of $7\left(e^{i s}+1\right)$ .

Find the second degree Taylor polynomial approximation of $\frac{1}{1+x^{2}}$ about x = 1.

Is the Taylor polynomial of degree 6 for $e^{-x^{2}}$ for x near 0 given by $1+\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}+\frac{x^{6}}{6 !}$ ?

Find $\lim _{x \rightarrow 0} \frac{\sin x}{4 x}$ using a Taylor approximation for sin x.

a) $1+2 x+4 x^{2}+8 x^{3}+\ldots$ is the Maclaurin series for what function? b)What is its radius and interval of convergence (excluding possible endpoints)? c)Use the Maclaurin series to determine $f^{\prime \prime \prime}(0)$ .

The hyperbolic cosine function is defined as follows: $f(x)=\cosh (x)=\frac{e^{x}+e^{-x}}{2}$ .Use the Taylor polynomial for $e^{x}$ near 0 to find the Taylor polynomial of degree 4 for $f(x)=4 \cosh (x)$ .

Find the first harmonic of the function $h(x)=$$\left\{\begin{array}{l}\pi \\0\end{array}\right.$ -\pi < 0< x\leq0 x\leq\pi .

Find the second harmonic of the function $f(x)=$$\left\{\begin{array}{c}-2 \\2\end{array}\right.$ -\pi< 0< x\leq0 x\leq\pi .

Use the formula for the Taylor polynomial approximation to the function $g(x)=e^{x}$ about $x_{0}=0$ to construct a polynomial approximation of degree 6 for $f(x)=e^{x^{2}}$ .Use the first four nonzero terms of this approximation to estimate the value of $e^{(0.3)^{2}}$ .Give your answer to 5 decimal places.

Since $f(x)=\ln x$ and $g(x)=e^{x}$ are inverse functions, we know that $e^{\ln (1+x)}=1+x$ for x > -1.Find the Taylor series for $e^{\ln (1+x)}$ using only up to the quadratic terms and show that the result is 1 + x.

Given the fact that the Taylor series about x = 0 for $e^{x}=1+\frac{x}{1 !}+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\cdots$ , is the Taylor series about x = 0 for $e^{x / 4}=1+\frac{x}{4}+\frac{x^{2}}{32}+\frac{x^{3}}{384}+\cdots$ ?

Show that the Taylor series about 0 for $f(x)=3 \sin (x)$ converges to $3 \sin (x)$ for all values of x by showing that the error $E_{n}(x) \rightarrow 0$ .

Find the first four non-zero terms of the Taylor series about zero for the function $f(x)=7(1+x)^{1 / 2}$ .Leave coefficients in fraction form.

The infinite series $x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\frac{x^{4}}{4}+\ldots+\frac{(-1)^{n-1} x^{n}}{n}+\ldots$ does not converge for $x=-1$ .What behavior does it exhibit? It does converge for $x=1$ .To what number does it appear to converge?