Chapter 12

Sum the series. $$\sum_{k=0}^{\infty} \frac{3 k}{k !} x^{3 k-1}$$

Let \(\sum a_{k} x^{k}\) be a series with radius of convergence \(r>0\) (a) Show that if the series is absolutely convergent at one endpoint of its interval of convergence, then it is absolutely convergent at the other endpoint. (b) Show that if the interval of convergence is \((-r . r)\) then the series is only conditionally convergent at \(r\)

Let \(r\) be a positive number. Show that \(a_{k}=r^{k} / k ! \rightarrow 0\) by considering the series \(\sum a_{k}\)

Set \(f(x)=\frac{e^{x}-1}{x}\) (a) Expand \(f(x)\) in a power series. (b) Integrate the series and show that .$$\sum_{n=1}^{\infty} \frac{n}{(n+1) !}=1$$

Let \(r>0\) be arbitrary. Give an example of a power series \(\sum a_{k} x^{k}\) with radius of convergence \(r\)

Show that if \(\sum a_{2}\) is absolutely convergent and \(\left|b_{k}\right| \leq\left|a_{k}\right|\) for at \(J k,\) then \(\sum b_{k}\) is absolutely convergent.

(a) Show that if \(\sum a_{\underline{f}}\) is absolutely convergent, then \(\sum a_{k}^{2}\) is convergent. (b) Show by means of an example that the converse of the result in part (a) is False.

Let \(P_{n}\) be the \(n\) th Taylor polynomial for the function $$f(x)=\ln (1+x)$$ Find the least integer \(n\) for which: (a) \(P_{n}(0.5)\) approximates in 1.5 within \(0.01 ;\) (b) \(P_{n}(0.3)\) approximates \(\ln 1.3\) within \(0.01 ;(c) P_{n}(1)\) approximates \(\ln 2\) within 0.001.

(a) How many terms of the series \(\sum_{k=1}^{\infty} \frac{1}{k^{4}}\) must you use to ensure that \(R_{n}\) is less than \(0.0001 ?\) (b) How large do you have to choose \(n\) to ensure that \(R_{n}\) is less than \(0.001 ?\) (c) Use the result of part (b) to estimate \(\sum_{t=1}^{\infty} \frac{1}{k^{-t}}\)

Set \(f(x)=x e^{x}\) (a) Expand \(f(x)\) in a power series. (b) Integrate the series and show that .$$\sum_{n=1}^{\infty} \frac{1}{n !(n-2)}=\frac{1}{2}$$

Find the integers \(p \geq 2\) for which \(\sum \frac{(k !)^{2}}{(p k) !}\) converges.

Find the interval of convergence of the series \(\sum s_{k} x^{k}\) where \(s_{k}\) is the \(k\) the partial sum of the series $$\sum_{n=1}^{\infty} \frac{1}{n}$$

Let \(P_{s}\) be the \(n\) th Taylor polynomial for the function $$f(x)=\sin x$$ Find the least integer \(n\) for which: (a) \(P_{n}(1)\) approximates sin 1 within 0.001; (b) \(P_{n}(2)\) approximates sin 2 within 0.001. (c) \(P_{n}(3)\) approximate \(\sin 3\) within 0.001.

Exercise 45 for the series \(\sum_{k=1}^{\infty} \frac{1}{k^{5}}\)

Deducc the differen:stion formulas $$ \frac{d}{d x}(\sinh x)=\cosh x, \quad \frac{d}{d x}(\cosh x)=\sinh x $$ from the expansions of \(\sinh x\) and \(\cosh x\) in powers of \(x\)

Let \(\sum_{i=1}^{\infty}(-1)^{k} a_{k}\) be an alternating series with the \(a_{k}\) forming a decreasing sequence of positive numbers. Show that the sequence of partial sums of odd index increases and is bounded above.

Show that, if \(\sum a_{k} x^{k}\) and \(\sum b_{k} x^{k}\) both converge to the same sum on some interval, then \(a_{k}=b_{k}\) for each \(k\)

Let \(\sum_{i=0}^{\infty} a_{k} x^{k}\) be a power series with radius of convergence \(r, r\) possibly infinite. (a) Giventhat \(\left|a_{k}\right|^{1 / k} \rightarrow \rho\) show that, if \(\rho: j\), then \(r=1 / \rho\) and, if \(\rho=0,\) then \(r=\infty\) (b) Given that \(\left|a_{k+1} / a_{k}\right| \rightarrow \lambda\) show that, if \(\lambda \neq 0\), then \(r=1 / \lambda\) and if \(\lambda=0,\) then \(r\) \(\infty\)

Set \(f(x)=e^{x}\) (a) Find the Taylor polynomial \(P_{n}\) for \(f\) of least degree that approximates \(e^{x}\) at \(x=\frac{1}{2}\) with four decimal place accuracy. Then use that polynomial to obtain an estimate of \(\sqrt{c}\). (b) Find the Taylor polynomial \(P_{n}\) for \(f\) of least degree that approximates \(e^{x}\) at \(x=-1\) with four decimal place accuracy. Then use that polynomial to obtain an estimate of \(1 / e\).

Take \(r>0\) and let the \(a_{k}\) be positive. Use the root test to show that, if \(\left(a_{k}\right)^{1 / k} \rightarrow \rho\) and \(\rho<1 / r,\) then \(\sum a_{i} r^{k}\) converges.

Complete the limit comparison test. Let: \(\sum a_{k}\) and \(\sum b_{z}\) be scrics with positive terms. Suppose that \(a_{k} / b_{k} \rightarrow 0\) (a) Show that if \(\sum b_{k}\) converges, then \(\sum a_{k}\) converges. (b) Show that if \(\sum a_{k}\) diverges, then \(\sum b_{i}\) diverges. (c) Show by cxanple that if \(\sum a_{k}\) converges, iben \(\sum b_{k}\) may converge or diverge. (d) Show by example that if \(\sum b_{k}\) diverges, fhen \(\sum a_{k}\) may converge or diverge. [Parts (c) and (d) explain why we slipulatcd \(L \Rightarrow 0\) in Theorem \(12.3 .7 .]\)

Let \(\sum a_{k} x^{k}\) be a power series with finite radius of convergence \(r\). Show that the power series \(\sum a_{k} x^{2 k}\) has radius of convergence \(\sqrt{r}\)

Form the series $$ a-\frac{1}{2} b+\frac{1}{3} a-\frac{1}{4} b+\frac{1}{3} a-\frac{1}{6} b+\dots $$ (a) Express this series in \(\sum\) notation. (b) For what positive values of \(a\) and \(b\) is this series absolutely convergent? conditionally convergent?

Set $$a_{k}=\left\\{\begin{array}{cl}\frac{1}{2^{6}} & \text { for odd } k \\\\\frac{1}{2^{k-2}} & \text { for even } k\end{array}\right.$$ The resulting series $$\sum_{k=1}^{\infty} a^{k}=\frac{1}{2}+1+\frac{1}{8}+\frac{1}{4}+\dots$$ is a rearrangement of the geometric series $$\sum_{k=0}^{\infty} \frac{1}{2^{k}}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\dots$$ (a) Use the root test to show that \(\sum a_{k}\) converges. (b) Show that the ratio test does not apply.

Sct \(f(x)=\cos x\) (a) Find the Taylor polynomial \(P_{n}\) for \(f\) of least degree that approximates \(\cos x\) at \(x=\pi / 30\) with three decimal place accuracy, Then use that polynomial to obtain an estimate of \(\cos (\pi / 30)\). (b) Find the Taylor polynomial \(P_{n}\) for \(f\) of least degree that approximates \(\cos x^{\circ}\) at \(x=9\) with four decimal place ac curacy. Then use that polynomial! : o obtain an estimate of \(\cos 9^{\circ}\)

Let \(\sum a_{t}\) and \(\sum b_{k}\) be series with positive terms. Suppose that \(a_{k} / b_{k} \rightarrow \infty\) (a) Show that if \(\sum b_{k}\) diverges, then \(\sum a_{k}\) diverges. (b) Show that if \(\sum a_{k}\) converges, then \(\sum b_{k}\) converges. (c) Show by example that if \(\sum a_{k}\) diverges, then \(\sum b_{k}\) may converge or diverge. (d) Show by example that if \(\sum b_{k}\) converges, then \(\sum a_{k}\) may converge or may div urgc.

Show that, if \(\epsilon > 0,\) then \(\left|k x^{k-1}\right| < (|x|+\epsilon)^{k}\) for all \(k\) sufficiently large.

Suppose that the function \(f\) has the power series representation \(f(x)=\sum_{k=0}^{\infty} n_{k} x^{k}\) (a) Show that if \(f\) is an even function, then \(a_{2 k+1}=0\) for all k. (b) Show that if \(f\) is an odd function, then \(a_{2 k}=0\) for all \(k\)

Let \(\sum a_{k}\) be is series with nonneyalive terms. (a) Show that if \(\sum a_{k}\) converges, then \(\sum a_{k}^{2}\) convery. (b) Give an example where \(\sum a_{k}^{2}\) converges and \(\sum a_{k}\) converges; give an example where \(\sum a_{k}^{2}\) converges but \(\sum a_{k}\) diverges.

Show that $$\cos x=\sum_{k=0}^{\infty} \frac{(-1)^{k}}{(2 k) !} x^{2 i} \quad \text { for all real } x$$ .

Suppose that the function \(f\) is infinitely differentiable on an oper: interval that contains \(0,\) and suppose that \(f^{\prime}(x)=\) \(-2 f(x)\) and \(f(0)=1 .\) Express \(f(x)\) as a power series in \(x\) What is the sum of this series?

Suppose that the function \(f\) is infinitely differentiable on an often interval that contains \(0,\) and suppose that \(f^{\prime \prime}(x)=\) \(-2 f(x)\) for all \(x\) and \(f(0)=0 . f^{\prime}(0)=1 .\) Express \(f(x)\) as a power series in \(x .\) What is the sum of this series?

Show that $$\sinh x=\sum_{k=0}^{\infty} \frac{1}{(2 k+1)} x^{2 k+1} \text { for all real } x$$

Let \(f\) be a continuous, positive, decreasing function on [1, \infty) for which \(\int_{1}^{\infty} f(x) d x\) converges. Then we know that the series \(\sum_{k=1}^{\infty} f(k)\) also converges. Show that $$0

Expand \(f(x) . f^{\prime}(x),\) and \(\int f(x) d x\) in power series (a) \(f(x)=x 2^{-1}\) (b) \(f(x)=x \arctan x\)

Show that $$\cosh x=\sum_{k=0}^{\infty} \frac{1}{(2 k) !} x^{2 k} \quad \text { for all real } x$$

Estimate within 0.001 by series expansion and check your result by carrying out the integration directly. $$\int_{0}^{1 / 2} x \ln (1-x) d x$$

Derive a series expansion in \(x\) for the function and specify the numbers \(x\) for which the expansion is valid. Take \(a>0\). $$f(x)=e^{2 x}$$

Estimate within 0.001 by series expansion and check your result by carrying out the integration directly. $$\int_{0}^{1} x \sin x d x$$

Derive a series expansion in \(x\) for the function and specify the numbers \(x\) for which the expansion is valid. Take \(a>0\). $$f(x) \equiv \sin a x$$

Let \(s_{n}\) be the \(n\) in partial sum of the harmonic serics, a series which you know diverges. (a) Show that $$\ln (n+1) < s_{n} < 1+\ln n$$ (b) Find the least integer \(n\) for which \(s_{n} > 100\).

Derive a series expansion in \(x\) for the function and specify the numbers \(x\) for which the expansion is valid. Take \(a>0\). $$f(x)=\cos a x$$

Show that $$ \begin{aligned} 0 &: \int_{0}^{2} e^{x^{2}} d x-\left[2+\frac{2^{3}}{3}+\frac{2^{5}}{5(2 !)}+\cdots+\frac{2^{2 n+1}}{(2 n+1) n !}\right] \\ & \cdot \frac{e^{4} 2^{2 n+1}}{(n+1) !} \end{aligned} $$

Derive a series expansion in \(x\) for the function and specify the numbers \(x\) for which the expansion is valid. Take \(a>0\). $$f(x)=\ln (1-a x)$$

Let \(p\) and \(q\) be polynomials with nonnegative cocfficients. Give necessary and sufficicnt conditions on \(p\) and \(q\) for the convergence of $$\sum \frac{p(k)}{q(k)}$$

Derive a series expansion in \(x\) for the function and specify the numbers \(x\) for which the expansion is valid. Take \(a>0\). $$f(x)=\ln (a+x)$$

Use the first two nonzero terms of \((12.6 .9)\) to estimate \(\ln 1.4\)

Verify the identity $$\frac{f^{(k)}(0)}{k !} x^{k}=\frac{1}{(k-1) !} \int_{0}^{x} f^{(k)}(t)(x-t)^{k-1} d t$$ $$-\frac{1}{k !} \int_{0}^{x} f^{(k+1)}(t)(x-t)^{k} d t$$ by using integration by parts on the second integral.

(a) Use a graphing utility to draw the graph of the function $$f(x)=\left\\{\begin{aligned} e^{-1 / x^{2}}, & x \neq 0 \\ 0, & x=0 \end{aligned}\right.$$ (b) Use L' Hòpial's rule to show that for every positive integer \(n\) $$\lim _{x \rightarrow 0} \frac{e^{-1 / x^{2}}}{x^{n}}=0$$ (c) Prove by induction that \(f^{(n)}(0)=0\) for all \(n \geq 1\) (d) What is the Taylor series of \(f ?\) (e) For what values of \(x\) dei: \(s\) s the Taylor serics of \(f\) actually converge to \(f(x) ?\)

Set \(f(x) \quad \cos x .\) Using a graphing utility or a CAS, draw a figure that gives the graph of \(f\) and the graphs of the Taylor polynomials \(P_{2}, P_{4}, P_{6}, P_{8}\).