Functions to power series Find power series representations centered at 0 for the following functions using known power series. Give the interval of convergence for the resulting series. f(x)=frac{3}{3+x}

Question
Series
Functions to power series Find power series representations centered at 0 for the following functions using known power series. Give the interval of convergence for the resulting series.
$$f(x)=\frac{3}{3+x}$$

2021-02-04
Consider the function $$f(x)=\frac{3}{3+x}$$
$$f(x)=\frac{3}{3+x}$$
$$=\frac{3}{3(1+\frac{x}{3})}$$
$$=\frac{1}{(1+\frac{x}{3})}$$
$$=(1+\frac{x}{3})^{-1}$$
Since $$(1+x)^{-1}=1-x+x^2-x^3+...+(-1)^nx^n+...$$ valid when |x|
$$f(x)=(1+\frac{x}{3})^{-1}$$
$$=1-\frac{x}{3}+(\frac{x}{3})^2-(\frac{x}{3})^3+...+(-1)^n(\frac{x}{3})^n+...$$ valid for $$|\frac{x}{3}|<1$$</span>
$$=1-\frac{x}{3}+\frac{x^2}{9}-\frac{x^3}{27}+...+(-1)^n(\frac{x}{3})^n+...$$ the expansion valid for |x|
Therefore, the power series of expansion is
$$f(x)=1-\frac{x}{3}+\frac{x^2}{9}-\frac{x^3}{27}+...+(-1)^n(\frac{x}{3})^n+...$$ or $$f(x)=\sum_{n=0}^\infty(-1)^n(\frac{x}{3})^n$$
for |x|
The expansion valid for |x|
Thus, the radius of convergence is R=3.

Relevant Questions

Functions to power series Find power series representations centered at 0 for the following functions using known power series. Give the interval of convergence for the resulting series.
$$f(x)=\ln\sqrt{4-x}$$
Differentiating and integrating power series Find the power series representation for g centered at 0 by differentiating or integrating the power series for ƒ (perhaps more than once). Give the interval of convergence for the resulting series.
$$g(x)=-\frac{1}{(1+x)^2}\text{ using }f(x)=\frac{1}{1+x}$$
Power series for derivatives
a. Differentiate the Taylor series centered at 0 for the following functions.
b. Identify the function represented by the differentiated series.
c. Give the interval of convergence of the power series for the derivative.
$$f(x)=\ln(1+x)$$
Find the power series representation for g centered at 0 by differentiating or integrating the power series for f(perhaps more than once). Give the interval of convergence for the resulting series.
$$g(x)=\frac{21}{(1-21x)^2},f(x)=\frac{1}{1-21x}$$
Find the power series representation for g centered at 0 by differentiating or integrating the power series for f(perhaps more than once). Give the interval of convergence for the resulting series.
$$\displaystyle{g{{\left({x}\right)}}}={\frac{{{21}}}{{{\left({1}-{21}{x}\right)}^{{2}}}}},{f{{\left({x}\right)}}}={\frac{{{1}}}{{{1}-{21}{x}}}}$$
Use the Geometric Series Test to help you find a power series representation of $$f(x)=\frac{x}{(2+x^3)}$$ centered at 0. Find the interval and radius of convergence.
Power series for derivatives
a. Differentiate the Taylor series centered at 0 for the following functions.
b. Identify the function represented by the differentiated series.
c. Give the interval of convergence of the power series for the derivative.
$$f(x)=\tan^{-1}x$$
Taylor series and interval of convergence
a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.
b. Write the power series using summation notation.
c. Determine the interval of convergence of the series.
$$f(x)=\log_3(x+1),a=0$$

A random sample of $$n_1 = 14$$ winter days in Denver gave a sample mean pollution index $$x_1 = 43$$.
Previous studies show that $$\sigma_1 = 19$$.
For Englewood (a suburb of Denver), a random sample of $$n_2 = 12$$ winter days gave a sample mean pollution index of $$x_2 = 37$$.
Previous studies show that $$\sigma_2 = 13$$.
Assume the pollution index is normally distributed in both Englewood and Denver.
(a) State the null and alternate hypotheses.
$$H_0:\mu_1=\mu_2.\mu_1>\mu_2$$
$$H_0:\mu_1<\mu_2.\mu_1=\mu_2$$
$$H_0:\mu_1=\mu_2.\mu_1<\mu_2$$
$$H_0:\mu_1=\mu_2.\mu_1\neq\mu_2$$
(b) What sampling distribution will you use? What assumptions are you making? NKS The Student's t. We assume that both population distributions are approximately normal with known standard deviations.
The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.
The standard normal. We assume that both population distributions are approximately normal with known standard deviations.
The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations.
(c) What is the value of the sample test statistic? Compute the corresponding z or t value as appropriate.
(Test the difference $$\mu_1 - \mu_2$$. Round your answer to two decimal places.) NKS (d) Find (or estimate) the P-value. (Round your answer to four decimal places.)
(e) Based on your answers in parts (i)−(iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level \alpha?
At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are not statistically significant.
(f) Interpret your conclusion in the context of the application.
Reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver. (g) Find a 99% confidence interval for
$$\mu_1 - \mu_2$$.
$$\displaystyle{f{{\left({x}\right)}}}={\ln{\sqrt{{{1}-{x}^{{5}}}}}}$$