# Calculus 2: Series questions and answers Recent questions in Series
Series
ANSWERED ### $${\sum_{{{x}={1}}}^{\infty}}\frac{1}{{8}^{n}}$$

Series
ANSWERED ### Simplify $$\displaystyle{\left({\sec{\theta}}-{\tan{\theta}}\right)}{\left({1}+{\sin{\theta}}\right)}$$

Series
ANSWERED ### $${\sum_{{{x}={1}}}^{\infty}}\frac{{{n}+{6}}}{{{n}^{2}+{12}{n}+{8}}}$$

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ANSWERED ### Tara dog bakery makes dog treats ultimate peanut butter and cheesy butter. Ultimate peanut butter requires 2 hours of kitchen time 5 pounds of peanut butter and 1 worker each batch of chessy butter requires 4 hours of kitchen time 1 pound peanut butter 1 worker the bakery has 40 kitchen hours each week 40 pounds of peanut butter and 12 workers if each batch of ultimate peanutbutter brings revenue of 20 dollars and each batch of cheesy butter brings revenue of 30 dollars in. How many batches of each flavor should the bakery make to maximize revenue show all equations graphing of feasible reigons corner points identified and calculations to show maximum revenue combinations.

Series
ANSWERED ### Suppose you take a dose of m mg of a particular medication once per day. Assume f equals the fraction of the medication that remains in your blood one day later. Just after taking another dose of medication on the second day, the amount of medication in your blood equals the sum of the second dose and the fraction of the first dose remaining in your blood, which is m+mf. Continuing in this fashion, the amount of medication in your blood just after your nth does is $$\displaystyle{A}_{{{n}}}={m}+{m}{f}+\ldots+{m}{f}^{{{n}-{1}}}$$. For the given values off and m, calculate $$\displaystyle{A}_{{{5}}},{A}_{{{10}}},{A}_{{{30}}}$$, and $$\displaystyle\lim_{{{n}\rightarrow\infty}}{A}_{{{n}}}$$. Interpret the meaning of the limit $$\displaystyle\lim_{{{n}\rightarrow\infty}}{A}_{{{n}}}$$. $$f=0.25,$$ $$m=200mg.$$

Series
ANSWERED ### Determine whether each of the given sets is a real linear space, if addition and multiplication by real scalars are defined in the usual way. For those that are not, tell which axioms fail to hold. The function are real-valued. All rational functions.

Series
ANSWERED ### Which of the following statements are​ true? i.If $$a_{n}\ \text{and}\ f(n)\ \text{satisfy the requirements of the integral test, then} \sum_{n=1}^{\infty} a_n = \int_1^{\infty} f(x)dx$$. ii. The series $$\sum_{n=1}^{\infty} \frac{1}{n^p}\ \text{converges if}\ p > 1\ \text{and diverges if}\ p \leq 1$$. iii. The integral test does not apply to divergent sequences.

Series
ANSWERED ### Find the radius of convergence, R, of the series. $$\sum_{n=1}^\infty\frac{(2x+9)^n}{n^2}$$ Find the interval, I, of convergence of the series.

Series
ANSWERED ### Use the Ratio Test to determine the convergence or divergence of the series. If the Ratio Test is inconclusive, determine the convergence or divergence of the series using other methods. $$\sum_{n=1}^\infty\frac{n^2}{(n+1)(n^2+2)}$$

Series
ANSWERED ### Determine whether the series $$\sum a_n$$ an converges or diverges: Use the Alternating Series Test. $$\sum_{n=2}^\infty(-1)^n\frac{n}{\ln(n)}$$

Series
ANSWERED ### Find the radius of convergence and interval of convergence of the series. $$\sum_{n=1}^\infty\frac{x^n}{n5^n}$$

Series
ANSWERED ### Determine if the following series converges or diverges. If it is a converging geometric or telescoping series, or ca be written as one, provide what the series converges to. $$\displaystyle{\sum_{{{k}={2}}}^{\infty}}{\frac{{{k}}}{{{k}^{{3}}-{17}}}}$$

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

Series
ANSWERED ### Determine the first four terms of the Maclaurin series for $$\sin 2x$$ (a) by using the definition of Maclaurin series. (b) by replacing x by 2x in the series for sin 2x. (c) by multiplying 2 by the series for $$\sin x$$ by the series for cos x, because sin $$2x = 2$$ $$\sin x \cos x$$

Series
ANSWERED ### 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. $$g(x)=-\frac{1}{(1+x)^2}\text{ using }f(x)=\frac{1}{1+x}$$

Series
ANSWERED ### Determine the radius and interval of convergence for each of the following power series. $$\sum_{n=0}^\infty\frac{2^n(x-3)^n}{\sqrt{n+3}}$$

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

Series
ANSWERED ### Write out the first few terms of each series to show how the series start. Then find out the sum of the series. $$\sum_{n=0}^\infty(\frac{5}{2^n}+\frac{1}{3^n})$$

Series
ANSWERED ### Consider the series $$\sum_{n=1}^\infty\frac{(-1)^n}{n^2}$$ a) Show the series converges or diverges using the alternating series test. b) Approximate the sum using the 4-th partial sum($$S_4$$) of the series. c) Calculate the maximum error between partial sum($$S_4$$) and the sum of the series using the remainder term portion of the alternating series test.
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