Calculus Examples of Geometric Series and More

I want to prove that
$\sum_{x = 1}^{\infty} \frac{\ln x}{x^{2}} \leq 1$

Ella Bradshaw 2022-01-22

How to prove that
$\sum_{n = 1}^{\infty} \frac{1}{4^{n} \cos^{2} \frac{x}{2^{n}}} = \frac{1}{\sin^{2} x} - \frac{1}{x^{2}}$

Kaydence Huff 2022-01-22

Find
$M = \sum_{n = 1}^{\infty} \tan^{- 1} \frac{2}{n^{2}}$

Wendy Gutierrez 2022-01-22

I want to speed up the convergence of a series involving rational expressions the expression is
$\sum_{x = 1}^{\infty} {(- 1)}^{x} \frac{- x^{2} - 2 x + 1}{x^{4} + 2 x^{2} + 1}$

Tori Hines 2022-01-22

Define the sequence ${[a_{n}]}_{n \geq 2}$ by
$a_{n} = \sqrt{1 + \sqrt{2 + \dots + \sqrt{n}}}$
It is well known $a_{n} \to l$ for a certain real number l. Show that
$lim_{n \to \infty} \sqrt{n} \cdot \sqrt[n]{l - a_{n}} = \frac{\sqrt{e}}{2}$

Jamarion Kerr 2022-01-22

Find the series:
$\sum_{n = 1}^{\infty} ({(1 + \frac{1}{n})}^{n} - e)$

Kasey Haley 2022-01-22

How to find the sum of this series?
$\sum_{k = 0}^{\infty} \frac{2^{k}}{(\begin{matrix} 2 k + 1 \\ k \end{matrix})}$

plorinosakl 2022-01-22

I have some difficulty to prove the following limit:
$lim_{N \to \infty} \sum_{k = 1}^{N} \frac{1}{k + N} = \ln (2)$

kuntungw3 2022-01-22

Evaluate the integral
$\int_{- π}^{π} {| \sum_{n = 1}^{\infty} \frac{1}{2^{n}} e^{\in x} |}^{2} d x$

Dayton Burnett 2022-01-21

Find the value of $\sum_{r = 1}^{4} \frac{1}{2 - a^{r}}$ where $a^{k} (k = 0, 1, 2, 3, 4, 5)$ are fifth roots of unity.

dublattm 2022-01-21

What is the closed form for $\sum_{n = - \infty}^{\infty}$ ?

Swebaceacichegh 2022-01-21

Given $a_{1} = 1, a_{n + 1} = a_{n} + \frac{1}{a_{n}}$ , find $lim_{n \to \infty} \frac{a_{n}}{n}$

derlingasmh 2022-01-21

Is it possible to write this in closed form:
$\sum_{k = 0}^{n} (\begin{matrix} n \\ k \end{matrix}) \log ((\begin{matrix} n \\ k \end{matrix}))$

Laney Spears 2022-01-21

A problem I saw recently is to compute
$lim_{n \to \infty} n^{2} ({(1 + \frac{1}{n + 1})}^{n + 1} - {(1 + \frac{1}{n})}^{n})$

Jessie Jenkins 2022-01-21

How could one prove that:
$\sum_{j = 2}^{\infty} \prod_{k = 1}^{j} \frac{2 k}{j + k - 1} = π$

Gavin Frye 2022-01-21

Prove that:
$x \log (x) = (\frac{x - 1}{x}) + \frac{3}{2!} {(\frac{x - 1}{x})}^{2} + \frac{11}{3!} {(\frac{x - 1}{x})}^{3} + \dots \frac{S_{n}}{n!} {(\frac{x - 1}{x})}^{n}$
Where $S_{n} =$ absolute Striling Numbers of first kind $(0, 1, 3, 11, 50, 274 \dots)$

Duncan Reed 2022-01-21

A series is given as follows
$\frac{1}{6} + \frac{5}{6 \cdot 12} + \frac{5 \cdot 8}{6 \cdot 12 \cdot 18} + \frac{5 \cdot 8 \cdot 11}{6 \cdot 12 \cdot 18 \cdot 24} + \dots$

William Montgomery 2022-01-21

Does the following series converge or diverge?
$\sum_{n = 1}^{\infty} \frac{1}{\sqrt{n} + \sqrt{n + 1}}$

Motalli21 2022-01-21

Prove
$\csc (x) = \sum_{k = - \infty}^{\infty} \frac{{(- 1)}^{k}}{x + k π}$

spiderifilms6e 2022-01-21

Prove that:
$\sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{{(2 n + 1)}^{2 m + 1}} = \frac{{(- 1)}^{m} E_{2 m} π^{2 m + 1}}{4^{m + 1} (2 m)!}$

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