Chapter 12

Determine whether the series converges or diverse. $$\sum \frac{2 k+1}{\sqrt{k^{3}+1}}$$

$$\text { Show that } \sum_{k=1}^{n} \frac{1}{\sqrt{k}} \geq \sqrt{n}$$

Use Taylor polynomials to estimate the following within 0.01. $$\sin 1$$

Determine whether the series converges or diverges. $$\sum \frac{\ln k}{k^{2}}$$

Use a graphing utility or CAS to evaluate the sum. $$\sum_{k=0}^{50} \frac{1}{4^{k}}$$

Find the interval of convergence. $$\sum\left(\frac{k}{k-1}\right) \frac{(x+2)^{k}}{2^{k}}$$

Determine whether the series converges or diverse. $$\sum \frac{2 k+1}{\sqrt{k^{3}+1}}$$

Suppose that only \(90 \%\) of the outstanding currency is recirculated into the economy: then \(90 \%\) of that is spent, and so on. Under this hypothesis, what is the long-term economic value of a dollar?

Consider the following sequence of steps. First, take the unit interval [0,1] and delete the open interval \(\left(\frac{1}{3} \cdot \frac{2}{3}\right) .\) Next, delete the two open intervals \(\left(\frac{1}{8}, \frac{2}{9}\right)\) and \(\left(\frac{7}{8}, \frac{8}{9}\right)\) from the two intervals that remain after the first step. For the third step, delete the middle thirds from the four intervals that remain after the second step. Continue on in this manner. What is the sum of the lengths of the intervals that have been deleted? The set that remains after all of the "middle thirds" have been de'cted is called line Cantor middle third set. Give some points that arc in the Cantor set.

Use Taylor polynomials to estimate the following within 0.01. $$\ln 1.2$$

Expand \(g(x)\) as indicated. $$g(x)=\sin ^{2} x \text { in powers of } x-\frac{1}{2} \pi$$.

Test these series for (a) absolute convergence, (b) conditional convergence. \(\sum \frac{\sin (\pi k / 2)}{k \sqrt{k}}\).

Use a graphing utility or CAS to evaluate the sum. $$\sum_{k=1}^{50} \frac{1}{k^{2}}$$

Find a power series representation for the improper integral. $$\int_{0}^{x} \frac{1-\cos t}{t^{2}} d t$$

Find the interval of convergence. $$\sum \frac{\ln k}{k}(x+1)^{k}$$

Determine whether the series converges or diverse. $$\sum \frac{1}{\sqrt{2 k(k+1)}}$$

Start with a square that has sides four units long. Join the midpoints of the sides of the square to form a second square inside the first. Then join the midpoints of the sides of the second square to form a third square, and so on. See the figure. Find the sum of the areas of the squares.

Determine whether the series converges or diverges. $$\sum \frac{2 \cdot 4 \cdots 2 k}{(2 k) !}$$

Test these series for (a) absolute convergence, (b) conditional convergence. \(\sum \frac{\sin (\pi k / 4)}{k^{2}}\).

Determine whether the series converges or diverse. $$\sum k c^{-k^{2}}$$

Use a graphing utility or CAS to evaluate the sum. $$\sum_{k=0}^{50} \frac{1}{k !}$$

Use Taylor polynomials to estimate the following within 0.01. $$\cos 1$$

Find the interval of convergence. $$\sum(-1)^{*} \frac{k^{2}}{(k+1) !}(x+3)^{k}$$

Expand \(g(x)\) as indicated. $$g(x)=\cos ^{2} x \quad \text { in powers of } x-\pi.$$

(a) Show that if the series \(\sum a_{k}\) converges and the series \(\sum b_{k}\) diverges, then the series \(\sum\left(a_{k}+b_{k}\right)\) diverges. (b) Give examples to show that if \(\sum a_{k}\) and \(\sum b_{k}\) both diverge, then each of the series \(\sum\left(a_{k}+b_{k}\right) \quad\) and \(\sum\left(a_{k}-b_{\varepsilon}\right)\) may converge or may diverge.

Expand \(g(x)\) as indicated. \(g(x)=(1+2 x)^{-4} \quad\) in powers of \(x-2\).

Test these series for (a) absolute convergence, (b) conditional convergence. \(\frac{1}{2}-\frac{1}{3}-\frac{1}{4}+\dots+\frac{1}{3 k+2}-\frac{1}{3 k+3}-\frac{1}{3 k+4}+\cdots\).

Determine whether the series converges or diverse. $$\sum k^{2} 2^{-k^{3}}$$

Use a graphing utility or CAS to evaluate the sum. $$\sum_{k=0}^{50}\left(\frac{2}{3}\right)^{k}$$

Find a power series representation for the improper integral. $$\int_{0}^{x} \frac{\sinh t}{t} d t$$

Use Taylor polynomials to estimate the following within 0.01. $$e^{0.8}$$

Find the interval of convergence. $$\sum \frac{k^{3}}{e^{k}}(x-4)^{k}$$

Let \(\sum_{i=0}^{\infty} a_{z}\) be a convergent series and let \(R_{o}=\sum_{j=n+1}^{\infty} a_{k} .\) Prove that \(R_{n} \rightarrow 0\) as \(n \rightarrow \infty\). Note that if \(s_{n}\) is the nth partial sum of the series, then \(\sum_{k=9}^{\infty} a_{k}=s_{n}+R_{m} ; R_{n}\) is called the remainder.

Estimate to within 0.01 by using series. $$\int_{0}^{i} e^{i t} d x$$

Use Taylor polynomials to estimate the following within 0.01. $$\sin 10^{\circ}$$

Determine whether the series converges or diverges. $$\sum \frac{k !(2 k) !}{(3 k) !}$$

Test these series for (a) absolute convergence, (b) conditional convergence. \(\frac{2 \cdot 3}{4 \cdot 5}-\frac{5 \cdot 6}{7 \cdot 8}+\dots+(-1)^{2} \frac{(3 k+2)(3 k+3)}{(3 k+4)(3 k+5)}+\cdots\).

Find the interval of convergence. $$\sum\left(1+\frac{1}{k}\right)^{k} x^{k}$$

Determine whether the series converges or diverse. $$\sum \frac{2+\sin k}{k^{2}}$$

(a) Prove that if \(\sum_{k=0}^{\infty} a_{k}\) is a convergent series with all terms nonzero, then \(\sum_{x=0}^{\infty}\left(1 / a_{k}\right)\) diverges. (b) Suppose that \(a_{k}>0\) for all \(k\) and \(\sum_{k=1}^{\infty} a_{k}\) diverges. Show by example that \(\sum_{k=0}^{\infty}\left(1 / a_{k}\right)\) may converge and it may diverge.

Expand \(g(x)\) as indicated. \(g(x)=(x-1)^{n} \quad\) in powers of \(x\).

Use Taylor polynomials to estimate the following within 0.01. $$\cos 6^{\circ}$$

Determine whether the series converges or diverges. $$\sum \frac{\ln k}{k^{5 / 4}}$$

Determine whether the series converges or diverse. $$\sum \frac{2+\cos k}{\sqrt{k+1}}$$

Find the interval of convergence. $$\sum \frac{(-1)^{k} a^{k}}{k^{2}}(x-a)^{k}$$

Show that $$ \sum_{x=1}^{\infty} \ln \left(\frac{k+1}{k}\right) \quad \text { diverges } $$ although $$ \ln \left(\frac{k}{k}-\frac{1}{k}\right) \rightarrow 0 $$

(a) Expand \(e^{x}\) in powers of \(x-a.\) (b) Use the expansion to show that \(e^{x_{1}+x_{2}}=e^{y_{1}} e^{x_{2}}.\) (c) Expand \(e^{-x}\) in powers of \(x-a.\)

Find the Lagrange form of the remainder \(R_{n}(x)\). $$f(x)=e^{2 x} ; \quad n=4$$

Estimate to within 0.01 by using series. $$\int_{0}^{1} \sin \sqrt{x} d x$$

Determine whether the series converges or diverges. $$\sum \frac{k^{2 / 2}}{k !}$$