Chapter 7

In Exercises \(1-6,\) find the first five terms of the sequence of partial sums. $$ 1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}+\cdot \cdot $$

In Exercises \(1-8,\) write the first five terms of the sequence. \(a_{n}=2^{n}\)

Use the Integral Test to determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{1}{n+1} $$

In Exercises 1 and 2 , state where the power series is centered. $$ \sum_{n=0}^{\infty} n x^{n} $$

In Exercises \(1-4,\) find a first-degree polynomial function \(P_{1}\) whose value and slope agree with the value and slope of \(f\) at \(x=c .\) Use a graphing utility to graph \(f\) and \(P_{1} .\) What is \(P_{1}\) called? $$ f(x)=\frac{4}{\sqrt{x}}, \quad c=1 $$

In Exercises \(1-10,\) use the definition to find the Taylor series (centered at \(c\) ) for the function. $$ f(x)=e^{2 x}, \quad c=0 $$

In Exercises \(1-16,\) determine the convergence or divergence of the series. \(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}\)

Find a geometric power series for the function, centered at 0 , (a) by the technique shown in Examples 1 and 2 and (b) by long division. $$ f(x)=\frac{1}{2-x} $$

Find the first five terms of the sequence of partial sums. $$ \frac{1}{2 \cdot 3}+\frac{2}{3 \cdot 4}+\frac{3}{4 \cdot 5}+\frac{4}{5 \cdot 6}+\frac{5}{6 \cdot 7}+\cdots $$

Write the first five terms of the sequence. \(a_{n}=\frac{3^{n}}{n !}\)

Use the Integral Test to determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{2}{3 n+5} $$

In Exercises 1 and 2 , state where the power series is centered. $$ \sum_{n=0}^{\infty} \frac{(-1)^{n}(x-\pi)^{2 n}}{(2 n) !} $$

In Exercises \(1-4,\) find a first-degree polynomial function \(P_{1}\) whose value and slope agree with the value and slope of \(f\) at \(x=c .\) Use a graphing utility to graph \(f\) and \(P_{1} .\) What is \(P_{1}\) called? $$ f(x)=\frac{4}{\sqrt[3]{x}}, \quad c=8 $$

Use the definition to find the Taylor series (centered at \(c\) ) for the function. $$ f(x)=e^{3 x}, \quad c=0 $$

Determine the convergence or divergence of the series. \(\sum_{n=1}^{\infty} \frac{(-1)^{n+1} n}{2 n-1}\)

Find a geometric power series for the function, centered at 0 , (a) by the technique shown in Examples 1 and 2 and (b) by long division. $$ f(x)=\frac{1}{1+x} $$

Find the first five terms of the sequence of partial sums. $$ 3-\frac{9}{2}+\frac{27}{4}-\frac{81}{8}+\frac{243}{16}-\cdots $$

Write the first five terms of the sequence. \(a_{n}=\left(-\frac{1}{2}\right)^{n}\)

Use the Integral Test to determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} e^{-n} $$

In Exercises \(3-6,\) find the radius of convergence of the power series. $$ \sum_{n=0}^{\infty}(-1)^{n} \frac{x^{n}}{n+1} $$

In Exercises \(1-4,\) find a first-degree polynomial function \(P_{1}\) whose value and slope agree with the value and slope of \(f\) at \(x=c .\) Use a graphing utility to graph \(f\) and \(P_{1} .\) What is \(P_{1}\) called? $$ f(x)=\sec x, \quad c=\frac{\pi}{4} $$

Use the definition to find the Taylor series (centered at \(c\) ) for the function. $$ f(x)=\cos x, \quad c=\frac{\pi}{4} $$

Determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n} n^{2}}{n^{2}+1} $$

Find a power series for the function, centered at \(c,\) and determine the interval of convergence. $$ f(x)=\frac{1}{2-x}, \quad c=5 $$

Find the first five terms of the sequence of partial sums. $$ \frac{1}{1}+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\frac{1}{9}+\frac{1}{11}+\cdot \cdot $$

Write the first five terms of the sequence. \(a_{n}=\frac{2 n}{n+3}\)

Use the Integral Test to determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} n e^{-n / 2} $$

In Exercises \(3-6,\) find the radius of convergence of the power series. $$ \sum_{n=0}^{\infty}(2 x)^{n} $$

In Exercises \(1-4,\) find a first-degree polynomial function \(P_{1}\) whose value and slope agree with the value and slope of \(f\) at \(x=c .\) Use a graphing utility to graph \(f\) and \(P_{1} .\) What is \(P_{1}\) called? $$ f(x)=\tan x, \quad c=\frac{\pi}{4} $$

Use the definition to find the Taylor series (centered at \(c\) ) for the function. $$ f(x)=\sin x, \quad c=\frac{\pi}{4} $$

Determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n}}{\ln (n+1)} $$

Find a power series for the function, centered at \(c,\) and determine the interval of convergence. $$ f(x)=\frac{4}{5-x}, \quad c=-2 $$

Find the first five terms of the sequence of partial sums. $$ \sum_{n=1}^{\infty} \frac{3}{2^{n-1}} $$

Write the first five terms of the sequence. \(a_{n}=\sin \frac{n \pi}{2}\)

Use the Integral Test to determine the convergence or divergence of the series. $$ \frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\frac{1}{9}+\frac{1}{11}+\cdots $$

In Exercises \(3-6,\) find the radius of convergence of the power series. $$ \sum_{n=1}^{\infty} \frac{(2 x)^{n}}{n^{2}} $$

Conjecture Consider the function \(f(x)=\cos x\) and its Maclaurin polynomials \(P_{2}, P_{4},\) and \(P_{6}\) (see Example 5 ). (a) Use a graphing utility to graph \(f\) and the indicated polynomial approximations. (b) Evaluate and compare the values of \(f^{(n)}(0)\) and \(P_{n}^{(n)}(0)\) for \(n=2,4,\) and 6 (c) Use the results in part (b) to make a conjecture about \(f^{(n)}(0)\) and \(P_{n}^{(n)}(0)\)

Use the definition to find the Taylor series (centered at \(c\) ) for the function. $$ f(x)=\ln x, \quad c=1 $$

Determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n}}{\sqrt{n}} $$

Find a power series for the function, centered at \(c,\) and determine the interval of convergence. $$ f(x)=\frac{3}{2 x-1}, \quad c=0 $$

Find the first five terms of the sequence of partial sums. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n !} $$

Write the first five terms of the sequence. \(a_{n}=(-1)^{n+1}\left(\frac{2}{n}\right)\)

Use the Integral Test to determine the convergence or divergence of the series. $$ \frac{\ln 2}{2}+\frac{\ln 3}{3}+\frac{\ln 4}{4}+\frac{\ln 5}{5}+\frac{\ln 6}{6}+\cdots $$

In Exercises \(3-6,\) find the radius of convergence of the power series. $$ \sum_{n=0}^{\infty} \frac{(2 n) ! x^{2 n}}{n !} $$

\mathrm{Conjecture } Consider the function \(f(x)=x^{2} e^{x}\) (a) Find the Maclaurin polynomials \(P_{2}, P_{3},\) and \(P_{4}\) for \(f\). (b) Use a graphing utility to graph \(f, P_{2}, P_{3},\) and \(P_{4}\). (c) Evaluate and compare the values of \(f^{(n)}(0)\) and \(P_{n}^{(n)}(0)\) for \(n=2,3,\) and 4 (d) Use the results in part (c) to make a conjecture about \(f^{(n)}(0)\) and \(P_{n}^{(n)}(0)\)

Use the definition to find the Taylor series (centered at \(c\) ) for the function. $$ f(x)=e^{x}, \quad c=1 $$

Determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1} n}{n^{2}+1} $$

Find a power series for the function, centered at \(c,\) and determine the interval of convergence. $$ f(x)=\frac{3}{2 x-1}, \quad c=2 $$

In Exercises \(7-14,\) verify that the infinite series diverges. $$ \sum_{n=0}^{\infty} 1000(1.055)^{n} $$

Write the first five terms of the sequence. \(a_{n}=\frac{(-1)^{n(n+1) / 2}}{n^{2}}\)