Calculus Examples of Geometric Series and More

What is the idea to sum this series of powers?
$\sum_{n = 1}^{+ \infty} (n^{2} + n + 1) x^{n - 1}$

Loretta Cannon 2022-02-27

Is the series
$\sum_{n = 1}^{\infty} \frac{1}{n} (\sum_{k = 1}^{n} \frac{1}{k} {(\frac{1}{2})}^{n - k})$

Bruce Partridge 2022-02-27

For what values of $a > 0$ is the equation
$\sum_{n = 1}^{\infty} \frac{2}{{(n - 0.5)}^{2} + a} = \frac{π}{\sqrt{a}}$
true ?

Nyah Conrad 2022-02-27

Study the behaviour of
$\sum_{n = 1}^{\infty} \frac{n + \log (n)}{{(n + \cos (n))}^{3}}$

sacateundisco8i3 2022-02-27

If $\frac{1}{200} \sum_{n = 1}^{399} \frac{5^{200}}{5^{n} + 5^{200}} = \frac{a}{b}$ , then find $| a - b |$

ymhonnolfq 2022-02-27

How to prove:
$π^{2} = 18 \sum_{n = 0}^{\infty} \frac{n! n!}{(2 n + 2)!}$

Bradlee Hooper 2022-02-27

How do you prove that
$\frac{10^{n} - 1}{10^{n} - 2} = 1 + \sum_{i = 0}^{\infty} 2^{i} \times 10^{- n (i + 1)}$

An object is moving along the curve = . When the object passess through the point (9,3), its x-coordinate increases at a rate of 2 m/s. How fast is the distance from the object to the origin changing at this instant?

Zaccagliaodi 2022-02-26

Evaluate:
$\sum_{n = 1}^{\infty} \frac{n^{2} + 1}{n \cdot 2^{n - 1}}$

Jupellodseple804 2022-02-26

I have to prove
$\frac{\sin (n θ)}{\sin (θ)} = 2^{n - 1} \prod_{k = 1}^{n - 1} (\cos (θ) - \cos (\frac{π k}{n}))$

OIPLZLqm6 2022-02-26

I want to know how to compute the closed form for the series
$\sum_{n = 0}^{\infty} \frac{(2 n)!}{{(n!)}^{2}} {(\frac{| A |}{2})}^{2 n}, | A | < 1,$

Arslan Lyons 2022-02-26

What is $S_{m} (n)$ if $S_{0} (n) = 1$ and $S_{m + 1} (n) = \sum_{k = 1}^{n} S_{m} (k)$ and m and n are integers?

SnuluddidaPalbx4 2022-02-26

Value of $\sum_{n = 0}^{1947} \frac{1}{2^{n} + \sqrt{2^{1947}}}$ ?

Carole Juarez 2022-02-26

I have to prove the following:
$\int_{0}^{1} \frac{x^{\frac{1}{3}}}{1 - x} \log \frac{1}{x} d x = \sum_{n = 0}^{\infty} \frac{9}{{(3 n + 4)}^{2}}$

ymhonnolfq 2022-02-26

What function does $\sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} (2 n)!}{(1 - 2 n) n!^{2} 4^{n}} \frac{x^{n}}{y^{2 n}}$ correspond to?

Myah Fuller 2022-02-26

Find convergence of:
$\sum_{n = 1}^{\infty} \frac{1}{{(n!)}^{\frac{2}{n}}}$

Nettie Potts 2022-02-26

How should I prove convergence using Cauchy sequences
$\sum_{k = 1}^{n} \frac{\sin (k^{3} + 1)}{(4 k + 1) (4 k + 5)}$

simnelsbl4 2022-02-26

Determine the convergence of
$\sum_{n = 1}^{+ \infty} \int_{0}^{\frac{1}{n}} \frac{\sin x}{1 + x} d x$

erycletrefeebr 2022-02-25

Find the sum:
$\sum_{k = 0}^{\infty} {(- 2)}^{k} \frac{k + 2}{k + 1} x^{k}$

Adrienne Rowe 2022-02-25

Does the sum $\sum_{n = 1}^{\infty} \frac{\sqrt{n!}}{(3 + \sqrt{1}) (3 + \sqrt{2}) \dots (3 + \sqrt{n})}$ converge or diverge?

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