Calculus Examples of Geometric Series and More

The series $2 [\frac{1}{3 x + 1} + \frac{1}{3 {(3 x + 1)}^{3}} + \frac{1}{5 {(3 x + 1)}^{5}} + \dots]$ is equal to?

Nicontio1 2021-12-23

I was asked to compute this series:
$\sum_{n = 1}^{\infty} \frac{1}{{(2 n - 1)}^{2}}$
but by using the fact that $\sum_{n = 1}^{\infty} \frac{1}{n^{2}} = \frac{π^{2}}{6}$

untchick04tm 2021-12-23

Find the Maclaurin series for $f (x) = \frac{x^{2}}{1 - 8 x^{8}}$

kerrum75 2021-12-22

Use the comparison or limit comparison test to decide if the following series converge.
$\sum_{n = 1}^{\infty} \frac{4 - \sin n}{2^{n} + 1}$
For each series which converges, give an approximation of its sum, together with an error estimate, as follows. First calculate the sum $s_{5}$ of the first 5 terms, then estimate the "tail" $\sum_{n = 6}^{\infty} a_{n}$ by comparing it with an appropriate improper integral or geometric series.

Jason Yuhas 2021-12-22

I have to find the sine series of $x^{3}$ using the the cosine series of $\frac{x^{2}}{2}$
$\frac{x^{2}}{2} = \frac{l^{2}}{6} + \frac{2 l^{2}}{π^{2}} [\sum \frac{{(- 1)}^{n}}{n^{2}} \cos (\frac{n π x}{l})]$

Mary Reyes 2021-12-22

I have to calculate the series
$\sum_{k = 2}^{\infty} (\frac{1}{k} - \frac{1}{k + 2})$
Using the definition:
$L = lim_{n \to \infty} S_{n} = lim_{n \to \infty} \sum_{k = 0}^{n} a^{k}$
Obviously $lim_{n \to \infty} (\frac{1}{n} - \frac{1}{n + 2}) = 0$ , but I don't think that this is the right way to calculate the value of the series.

Michael Maggard 2021-12-22

Is there a way to rewrite a series like $\sum_{n = 1}^{\infty}$ as a power series? I know its

Bobbie Comstock 2021-12-22

Determined the convergence of this alternating series. The series is $\sum_{i = 1}^{\infty} 2 (\frac{- 1}{4})^{(n - 1)}$

eiraszero11cu 2021-12-20

Which of the answers is an arithmetic sequence?
a) 2, 4, 8, 16, 32
b) 3, 6, 9, 15, 24
c) 2, 5, 7, 12, 19
d). 6, 13, 20, 27, 34

rheisf 2021-12-18

Select the FIRST correct reason why the given deries converges.
A. Convergent geometric series
B. Convergent p series
C. Comparison (of Limit Comparison) with a geometric or p series
D. Converges by alternating series test
1. $\sum_{n = 1}^{\infty} \frac{\sin^{2} (3 n)}{n^{2}}$
2. $\sum_{n = 1}^{\infty} \frac{{(- 1)}^{n} \ln (e^{n})}{n^{4} \cos (n π)}$
3. $\sum_{n = 1}^{\infty} \frac{\cos (n π)}{\ln (4 n)}$
4. $\sum_{n = 1}^{\infty} \frac{2 {(4)}^{n}}{6^{2 n}}$
5. $\sum_{n = 1}^{\infty} \frac{{(- 1)}^{n}}{2 n + 6}$
6. $\sum_{n = 1}^{\infty} {(- 1)}^{n} \frac{\sqrt{n}}{n + 4}$

Adela Brown 2021-12-16

Which statement is true for the following series?
$\sum_{k = 1}^{\infty} \frac{{(- 1)}^{k}}{5 k + 6}$
1. Cannot be determined
2. Divergent
3. Conditionally convergent
4. Absolutely convergent

Holly Guerrero 2021-12-14

The series $\sum_{n = 1}^{\infty} n^{k} r^{n}$ coverges when $r \in (0, 1)$ and diverges when $r > 1$ . This is true regardless of the value of the constant k. When $r = 1$ the series is a p-series. It converges if $k < - 1$ and diverges otherwise Each of the series below can be compared to a series of the form $\sum_{n = 1}^{\infty} n^{k} r^{n}$ . For each series determine the best value of r and decide whether the series converges.
I am stuck on this question $\sum_{n = 1}^{\infty} {(\frac{3 n^{2} + 4 n + 2^{- 2 n}}{7^{n + 2} + 4 n + 5 \sqrt{n}})}^{2}$
I tried using $\frac{2^{- 2 n}}{7^{n + 2}}$ and ended up with a value of r of $\frac{1}{4 \cdot 7}$ but this didn't work.

Priscilla Johnston 2021-12-13

How do you find a formula for the sum n terms $\sum_{i = 1}^{n} (1 + \frac{i}{n}) (\frac{2}{n})$ and then find the limit as $n \to \infty$

idiopatia0f 2021-12-13

Given a data set consisting of 33 unique whole number observations, its five - number summary is [ 12,24,38,51,64] How many observations are less than 38?

Judith McQueen 2021-12-12

How to find the Maclaurin series for $e^{x}$ ?

NompsypeFeplk 2021-11-21

How to multiply two summations together?

dictetzqh 2021-11-17

Which equation could be used to calculate the sum of the geometric series?
$\frac{1}{3} + \frac{2}{9} + \frac{4}{27} + \frac{8}{21} + \frac{16}{243}$

ArcactCatmedeq8 2021-11-17

Use the definition of Taylor series to find the Taylor series, centered at c for the function. $f (x) = \ln x, c = 1$

angepepagiortzb 2021-11-15

Use the summation formulas to rewrite the expression withous summatiom notation.
$\sum_{i = 1}^{n} \frac{4 i + 5}{n^{2}}$
Use the result to find the sums for n=10, 100.

Boduszewox6 2021-11-14

Write an equivalent series with the index of summation beginning at = 1.
$\sum_{n = 0}^{\infty} {(- 1)}^{n} + 1 (n + 1) x^{n}$

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