Calculus Examples of Geometric Series and More

Showing that:
$lim_{n \to \infty} \frac{\sqrt[n]{n!}}{n} = \frac{1}{e}$

Shirley Thompson 2022-01-16

Whats

eliaskidszs 2022-01-16

I encountered while learning about definite integrals:
$\sum_{k = 1}^{n} k^{2} = \frac{n (n + 1) (2 n + 1)}{6}$

Talamancoeb 2022-01-07

I have to solve -
$\sum_{i = 1}^{\infty} {(\frac{5}{12})}^{i}$
- geometric series?

jubateee 2022-01-07

Test the convergence of the following series:
$\frac{\sqrt{n + 1} - 1}{{(n + 2)}^{3} - 1} + \dots + \infty$

William Cleghorn 2022-01-06

What is the appropriate way to simplify such an expression. i am unsure of how to use the series i know to apply to this situation
$\sum_{L = 0}^{M} s^{L} L^{2}$

eliaskidszs 2022-01-06

Using power series representation, calculate
$\sum_{n = 1}^{\infty} \frac{n 2^{n}}{3^{n}}$

Talamancoeb 2022-01-05

I really can't remember (if I have ever known this): which series is this and how to demonstrate its solution?
$\sum_{i = 1}^{n} i = \frac{n (n + 1)}{2}$

burkinaval1b 2022-01-04

Having trouble with this infinite series and deciding whether it converges or diverges.
The series:
$\sum_{n = 1}^{\infty} n {(\frac{1}{2 i})}^{n}$

Joan Thompson 2022-01-04

Consider the following infinite series:
$\sum_{k = 0}^{\infty} \frac{2^{k}}{1 + \frac{1}{x^{2 k}}}$

zagonek34 2022-01-03

I fell upon this very interesting series in a textbook:
$S_{n} = \sum_{p = 1}^{n} \frac{1}{n + p}$

kloseyq 2022-01-02

How do I evaluate the power series
$\sum_{n = 1}^{\infty} \frac{n}{9^{n}}$
without using the formula for infinite geometric series?

Juan Hewlett 2022-01-02

Show that the series converges.
$\sum_{n = 1}^{\infty} \frac{n!^{n}}{n^{n^{2}}}$

Anne Wacker 2022-01-01

Find the complex form of the Fourier Series expansion of
$f (x) = \cos a x, - π < x < π$

kuvitia9f 2022-01-01

Looking to prove that the following series converges conditionally
$\sum_{n = 1}^{\infty} \frac{{(- 1)}^{n + 1} {(1 + n)}^{\frac{1}{n}}}{n}$

Irvin Dukes 2021-12-26

Use the comparison or limit comparison test to decide if the following series converge.
$\sum_{n = 1}^{\infty} \frac{4 - \sin n}{n^{2} + 1}$
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.

Daniell Phillips 2021-12-26

Using Taylor expansion check if series :
$\sum_{n = 1}^{\infty} \sqrt{n} (\arctan (n + 1) - \arctan n)$
converges.

namenerk 2021-12-25

If the Taylor Series of $\ln (x)$ is known:
$\ln (x) = (x - 1) - \frac{1}{2} {(x - 1)}^{2} + \frac{1}{3} {(x - 1)}^{2} - \frac{1}{4} {(x - 1)}^{4} + \frac{1}{5} {(x - 1)}^{5} - \dots$
Can one find the Taylor series of
$f (x) = \frac{x}{1 - x^{2}}$
by manipulating the Taylor series of $\ln (x)$ ?

Teddy Dillard 2021-12-25

I am having difficulty finding if this series converges or diverges:
$\sum_{n = 0}^{\infty} \frac{{(- 2)}^{n + 1}}{n^{n - 1}}$
I am unsure of which test to use. At first, I thought I should use alternating series test, but I am unable to manipulate the numerator to ${(- 1)}^{n + 1}$ . I thought I may be able to use root test, but I have no clue how to manipulate the series to do that.
What series test would I use, and how?

compagnia04 2021-12-25

The series is,
$\sum_{n = 9}^{\infty} \frac{1}{n (n - 1)}$

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