Calculus Examples of Geometric Series and More

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 $A_{n} = m + m f + \dots + m f^{n - 1}$ . For the given values off and m, calculate $A_{5}, A_{10}, A_{30}$ , and $lim_{n \to \infty} A_{n}$ . Interpret the meaning of the limit $lim_{n \to \infty} A_{n}$ . $f = 0.25,$ $m = 200 m g .$

Lewis Harvey 2021-07-02

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.

Rui Baldwin 2021-03-18

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.
$\sum_{n = 1}^{\infty} \frac{n^{2}}{(n + 1) (n^{2} + 2)}$

Cheyanne Leigh 2021-03-18

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

Cheyanne Leigh 2021-03-12

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

Bergen 2021-03-11

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.
$\sum_{k = 2}^{\infty} \frac{k}{k^{3} - 17}$

Wribreeminsl 2021-03-11

Find the radius of convergence and interval of convergence of the series.
$\sum_{n = 1}^{\infty} \frac{x^{n}}{n 5^{n}}$

banganX 2021-03-11

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.
$f (x) = \log_{3} (x + 1), a = 0$

Kye 2021-03-09

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

Kyran Hudson 2021-03-08

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

Chesley 2021-03-08

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

Mylo O'Moore 2021-03-08

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

opatovaL 2021-03-07

Use the Root Test to determine the convergence or divergence of the series.
$\sum_{n = 1}^{\infty} (\frac{n}{500})^{n}$

Zoe Oneal 2021-03-07

Determine whether the series converges or diverges.
$\sum_{n = 2}^{\infty} \frac{1}{n \ln n}$

OlmekinjP 2021-03-07

Determine the radius of convergence and the interval of convergence for each power series.
$\sum_{n = 0}^{\infty} \sqrt{n} (x - 1)^{n}$

DofotheroU 2021-03-07

Use the binomial series to find the Maclaurin series for the function $f (x) = \sqrt{1 + x^{3}}$

Kyran Hudson 2021-03-07

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.

banganX 2021-03-06

Verify that the infinite series converges.
$\sum_{n = 0}^{\infty} {(- 0.2)}^{n} = 1 - 0.2 + 0.04 - 0.008 + \dots$

rocedwrp 2021-03-06

Find the interval of convergence of the power series.
$\sum_{n = 0}^{\infty} \frac{x^{5 n}}{n!}$

iohanetc 2021-03-06

Consider the following series.
$\sum_{n = 1}^{\infty} \frac{\sqrt{n} + 4}{n^{2}}$
The series is equivalent to the sum of two p-series. Find the value of p for each series.
Determine whether the series is convergent or divergent.

The majority of questions about series calculus that you will find online will have odd solutions that do not really provide explanations or the answers that would help one understand how things work. We have taken a different approach by offering series calculus practice questions that are connected with initial equations that must be solved. It is once again about estimation and analysis as the examples include the building of the bridges and calculation of the power or energy necessary to implement the safest solution by estimating the best one. By offering clear examples, we help you learn things easier!

Calculus 2

Calculus Series: Understanding Geometric Examples