# 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

Question
Series
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.
$$f(x)=\log_3(x+1),a=0$$

2021-03-12

A) The Taylor series for a function f(x) continuously differentiable at $$x=a$$ is given by,
$$f(x)=f(a)+\frac{(x-a)}{1!}f'(a)+\frac{(x-a)^2}{2!}f''(a)+\frac{(x-a)^3}{3!}f'''(a)...$$
Here
$$f(x)=\log_3(x+1)$$
$$a=0,$$
Substitute the values,
$$f(x)=f(a)+\frac{(x-a)}{1!}f'(a)+\frac{(x-a)^2}{2!}f''(a)+\frac{(x-a)^3}{3!}f'''(a)...$$
$$\log_3(x+1)=\log_3(x+1)+\frac{x}{1!}(\frac{1}{(x+1)\log3})+\frac{x^2}{2\times1}(\frac{-1}{(x+1)^2\log3})+\frac{x^3}{3\times2\times1}[-(\frac{-2}{(x+1)^3\log3})]...$$
$$=\frac{\log(x+1)}{\log(3)}+\frac{x}{(x+1)\log3}-\frac{x^2}{2(x+1)^2\log3}+\frac{x^3}{3(x+1)^3\log3}...$$
The first four terms of the Taylor's expansion of $$f(x)=\log_3(x+1)$$ are respectively,
$$\frac{\log(x+1)}{\log_3},\frac{x}{(x+1)\log3},\frac{x^2}{2(x+1)^2\log3},\frac{x^3}{3(x+1)^3\log3}$$
B) The power series for the above Taylor's expansion is given by,
$$\log_3(x+1)=\sum_{n=0}^\infty\left\{\frac{(x-a)^n}{n!}\cdot f^n(a)\right\}$$
Re-arrange the series,
$$\log_3(x+1)=\frac{x}{(x+1)\log(3)}+\frac{x}{(x+1)\log(3)}-\frac{1}{\log(3)}[\frac{x^2}{2(x+1)^2}-\frac{x^3}{3(x+1)^3}+\frac{x^4}{4(x+1)^4}-\frac{x^5}{5(x+1)^5}+\frac{x^6}{6(x+1)^6}...]$$
$$=\frac{x}{(x+1)\log(3)}+\frac{x}{(x+1)\log(3)}-\frac{1}{\log(3)}\sum_{n=2}^\infty[\frac{x^n}{n(x+1)^n}\cdot(-1)^n]$$
The series will be converging if the term $$\sum_{n=2}^\infty[\frac{x^n}{n(x+1)^n}\cdot(-1)^n]$$ is converging,
Let $$a_n=(-1)^n\cdot\frac{x^n}{n(x+1)^n}$$
Apply the Ratio Test,
$$L=\lim_{n\rightarrow\infty}|\frac{a_{n+1}}{a_n}|$$
$$=\lim_{n\rightarrow\infty}|\frac{[\frac{x^{n+1}}{(n+1)(x+1)^{n+1}}\cdot(-1)^{n+1}]}{[\frac{x^n}{n(x+1)^n}\cdot(-1)^n]}|$$
$$=\lim_{n\rightarrow\infty}|\frac{-xn}{(x+1)(n+1)}|$$
$$=|\frac{x}{x+1}|\cdot\lim_{n\rightarrow\infty}|\frac{1}{1+\frac1n}|$$
$$=|\frac{x}{x+1}|$$
The series is converging if,
$$L<1$$
$$|\frac{x}{x+1}|<1$$
$$-1<\frac{x}{x+1}<1$$
The term $$\frac{x}{x+1}<1$$ is true for any real value of $$x \in R,$$
Also,
$$\frac{x}{x+1}<1$$
$$x>-x-1$$
$$2x>1$$
$$x>\frac{-1}{2}$$
Thus, the interval of convergence for the given power series is $$(\frac{-1}{2},\infty)$$

### Relevant Questions

Taylor series
a. Use the definition of a Taylor 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.
$$f(x)=2^x,a=1$$
Write out the first three nonzero terms of the Taylor series for the following functions centered at the given point a. Then write the series using summation notation.
$$f(x)=\tan^{-1}4x,a=0$$
Taylor series Write out the first three nonzero terms of the Taylor series for the following functions centered at the given point a. Then write the series using summation notation.
$$\displaystyle{f{{\left({x}\right)}}}={\text{cosh}{{\left({2}{x}-{2}\right)}}},{a}={1}$$
Any method
a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. You do not need to use the definition of the Taylor series coefficients.
$$f(x)=x^2\cos x^2$$
Binomial series
a. Find the first four nonzero terms of the binomial series centered at 0 for the given function.
b. Use the first four terms of the series to approximate the given quantity.
$$f(x)=(1+x)^{\frac{2}{3}}$$, approximate $$(1.02)^{\frac{2}{3}}$$

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
$$\mu_1 - \mu_2$$.
lower limit
upper limit
(h) Explain the meaning of the confidence interval in the context of the problem.
Because the interval contains only positive numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, we can not say that the mean population pollution index for Englewood is different than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains only negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is less than that of Denver.
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.
$$f(x)=\ln(1+x)$$
a) Find the Maclaurin series for the function
$$f(x)=\frac11+x$$
b) Use differentiation of power series and the result of part a) to find the Maclaurin series for the function
$$g(x)=\frac{1}{(x+1)^2}$$
c) Use differentiation of power series and the result of part b) to find the Maclaurin series for the function
$$h(x)=\frac{1}{(x+1)^3}$$
d) Find the sum of the series
$$\sum_{n=3}^\infty \frac{n(n-1)}{2n}$$
This is a Taylor series problem, I understand parts a - c but I do not understand how to do part d where the answer is $$\frac72$$
Find the Maclaurin series for using the definition of a Maclaurin series. [Assume that has a power series expansion.Do not show that Rn(x) tends to 0.] Also find the associated radius of convergence. $$f(x)=(1-x)^{-2}$$