91Ó°ÊÓ

Explain why the series is not a power series in$x-x_0$.Then use the ratio test for absolute convergence to find the values of $x$for which the given series converge role="math" localid="1649522407402" $\underset{k=1}{\overset{∞}{∑}}\frac{{\left(-1\right)}^k}{k}{\left(\frac{x}{x-1}\right)}^k$.

Use Theorem 8.12 and the results from Exercises 41–50 to find series equal to the definite integrals in Exercises 51–60.

${∫}_{0.5}^1\ln (4+{\mathrm{x}}^2)\mathrm{dx}$

Find the Maclaurin series for the functions in Exercises 51–60

by substituting into a known Maclaurin series. Also, give the

interval of convergence for the series.

$\frac{x}{9-x^2}$

In Exercises 49–56 find the Taylor series for the specified function and the given value of $x_0$.

56.$\sin 3x,\frac{\mathrm{Ï€}}{6}$

Explain why the series is not a power series in$x-x_0$.Then use the ratio test for absolute convergence to find the values of $x$ for which the given series converge role="math" localid="1649525417145" $\underset{k=1}{\overset{∞}{∑}}\frac{1}{k^3}{\left(\frac{x+2}{x-3}\right)}^k$

Show that if $p$is a positive integer, then the binomial series for $f\left(x\right)={(1+x)}^p$is a polynomial.

Use Theorem 8.12 and the results from Exercises 41–50 to find series equal to the definite integrals in Exercises 51–60.

${∫}_1^2{\mathrm{x}}^2\sin \left(5{\mathrm{x}}^2\right)\mathrm{dx}$

Find the Maclaurin series for the functions in Exercises 51–60

by substituting into a known Maclaurin series. Also, give the

interval of convergence for the series.

Explain why the series is not a power series in$x-x_0$ .Then use the ratio test for absolute convergence to find the values of $x$for which the given series convergerole="math" localid="1649526903036" $\underset{k=1}{\overset{∞}{∑}}\frac{{\left(-1\right)}^{k}k!}{k^2}{\left(\frac{3x}{x-2}\right)}^k$

Use Theorem 8.12 and the results from Exercises 41–50 to find series equal to the definite integrals in Exercises 51–60.

${∫}_{0.5}^2{\mathrm{x}}^3\cos \left(\frac{\mathrm{x}}{2}\right)\mathrm{dx}$