Calculus Examples of Geometric Series and More

What tools would you gladly recommend me for computing precisely the limit below? Maybe a starting point?
$lim_{n \to \infty} \frac{1}{\log (n)} \sum_{k = 1}^{n} \frac{\cos (\sin (2 π \log (k)))}{k}$

pozicijombx 2022-01-23

How do I find:
$\prod_{n = 1}^{\infty} (1 + \frac{1}{π^{2} n^{2}})$

Miguel Davenport 2022-01-23

For $x > 0$ and $y > x + 1$ , how do we prove that
$\sum_{n = 1}^{\infty} \frac{x (x + 1) \dots (x + n - 1)}{y (y + 1) \dots (y + n - 1)} = \frac{x}{y - x - 1}$

siutulysr5 2022-01-23

Evaluate
$\sum_{n, k} \frac{1}{(\begin{matrix} n \\ k \end{matrix})}$
where the summation ranges over all positive integers n, k with $1 < k < n - 1$

Nylah Church 2022-01-23

How to calculate $\int_{0}^{1} x^{x} d x$ using series? I read from a book that
$\int_{0}^{1} x^{x} d x = 1 - \frac{1}{2^{2}} + \frac{1}{3^{3}} + \dots + {(- 1)}^{n} \frac{1}{{(n + 1)}^{n + 1}} + \dots$

Carly Shannon 2022-01-23

For $a \in R$ , show that $\sum_{k \geq 0} \frac{1}{k^{2} + k - α} = \frac{π}{\sqrt{4 α + 1}} \tan (\frac{1}{2} π \sqrt{4 α + 1})$

Marquis Neal 2022-01-23

How to prove that
$\sum_{k \geq 1} \frac{1}{2^{k} k^{2}} = \frac{π^{2}}{12} - \frac{1}{2} {\log (2)}^{2}$

William Montgomery 2022-01-23

I want to try and evaluate this interesting sum:
$\sum_{n = 1}^{\infty} \frac{1}{Γ (n + s)}$
where $0 \leq s < q$

Jaeden Frederick 2022-01-23

How to show that that $\sum_{n = 1}^{\infty} (\frac{1}{3 n - 1} + \frac{1}{3 n - 2} - \frac{2}{3 n}) = \ln (3)$

Octavio Miller 2022-01-22

Evaluate
$\sum_{n = 1}^{\infty} \frac{1}{n^{2} 2^{n}}$

Jacquelyn Sanders 2022-01-22

How to compute
$\prod_{n = 2}^{\infty} \frac{n^{3} - 1}{n^{3} + 1}$

Shamar Padilla 2022-01-22

Does
$\sum_{n = 1}^{\infty} \frac{1}{n^{1 + \frac{1}{n}}}$
converge? If yes, to where?

maliaseth0 2022-01-22

Let x be any positive real number, and define a sequence $[a_{n}]$ by
$a_{n} = \frac{[x] + [2 x] + [3 x] + \dots + [n x]}{n^{2}}$
where $[x]$ is the largest integer less than or equal to x. Prove that $lim_{n \to \infty} a_{n} = \frac{x}{2}$

Branden Valentine 2022-01-22

Theres

Sereinserenormg 2022-01-22

Calculate the sun of the infinite series:
$\sum_{n = 0}^{\infty} \frac{n}{4^{n}}$

Elsa Barron 2022-01-22

Show that the following infinite series
$\sum_{i = 1}^{\infty} \frac{f (n)}{n^{2}}$

Jamarion Kerr 2022-01-22

Need to prove the sequence:
$a_{n} = 1 + \frac{1}{2^{2}} + \frac{1}{3^{2}} + \dots + \frac{1}{n^{2}}$ converges.

licencegpopc 2022-01-22

How to sum this series for $\frac{π}{2}$ directly?
$\frac{π}{2} = \sum_{k = 0}^{\infty} \frac{k!}{(2 k + 1)!}$

mairie0708zl 2022-01-22

Prove that :
$\sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{{(2 k + 1)}^{3}} = \frac{π^{3}}{32}$

spiderifilms6e 2022-01-22

Proving that $\frac{π^{3}}{32} = 1 - \sum_{k = 1}^{\infty} \frac{2 k (2 k + 1) ζ (2 k + 2)}{4^{2 k + 2}}$

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