Detailed Explanations for Calculus 2 Equations

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
$lim_{n \to \infty} \sqrt{n} \cdot \sqrt[n]{l - a_{n}} = \frac{\sqrt{e}}{2}$

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.

absolutestylehc 2022-01-21

Given
$1 \cdot 2^{1} + 2 \cdot 2^{2} + 3 \cdot 2^{3} + 4 \cdot 2^{4} + \dots + d \cdot 2^{d} = \sum_{r = 1}^{d} r \cdot 2^{r}$
how can we infer to the following solution?
$2 (d - 1) \cdot 2^{d} + 2$

stropa0u 2022-01-21

Prove the following inequality for every $n \geq 1$ :
$\sum_{k = 1}^{n} \frac{1}{k^{2} + 3 k + 1} \leq \frac{13}{20}$

kuntungw3 2022-01-21

Find $\sum_{n = 0}^{\infty} \frac{2^{n}}{3^{2^{n} - 1} + 1}$

Maximus George 2022-01-21

Evaluate $\sum_{k = 1}^{\infty} \frac{k^{2}}{(k - 1)!}$

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

When you are dealing with any Calculus 2 homework, it is vital to have a look at the various questions and answers that will help you see whether you are correct in your approach to finding solutions. Even if you are dealing with analytical aspects of Calculus 2, it will be helpful as you are looking at provided equations and learn how the answers relate to original questions and problems specified.

Do not be afraid to take a look at the basic integration and related application if Calculus 2 does not sound clear or start with the Calculus 1 first.

Calculus 2

Calculus 2 Equations: Expert Solutions and More