Calculus Examples of Geometric Series and More

Prove that:
$\sum_{k = 1}^{\infty} \frac{k! \sin (k)}{k^{k}}$
convergence or divergence?

Zachery Fellows 2022-02-25

With $γ$ being the Euler Mascheroni constant, this series is well known:
$1 - \sum_{n = 2}^{\infty} \frac{ζ (n) - 1}{n} = γ$ (1)
The following series involving $ζ (2 n + 1)$ also seems to converge to the same value, albeit slower:
$1 - \sum_{n = 1}^{\infty} \frac{ζ (2 n + 1)}{(n + 1) (2 n + 1)} = γ$ (2)
Is there a way to derive (2) from (1) ?

Cormac Gilbert 2022-02-25

So I have got this practice problem
$\sum_{k = 1}^{n} \frac{{(- 1)}^{k} \cdot 2^{2 k}}{3}$

Brittney Cuevas 2022-02-25

Simplify the power series
$\sum_{n = 0}^{\infty} \frac{(2 n)!}{n! (n + 1)!} \cdot p^{n} {(1 - p)}^{n}$

Ivor Schofield 2022-02-25

I would like to prove (or disprove) the following limit:
$lim_{n \to \infty} \frac{c}{n} \sum_{j = 1}^{n} {(1 - \frac{b c}{n})}^{2 (n - j)} = \frac{1}{2 b} (1 - e^{- 2 b c})$
for $b, c > 0$

embeucadaoo8 2022-02-25

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

Rome Frazier 2022-02-25

Proving that:
$\sum_{n = 1}^{99} \frac{\sqrt{n + 1} - \sqrt{n}}{2 n + 1} < \frac{9}{20}$

Tajzler6ek 2022-02-25

Convergence of $\sum_{n = 1}^{\infty} \frac{e^{- x n} \ln x}{x^{2} + n}$ if $x > 0$

Ingrid Senior 2022-02-25

Proof that Dirichlet series
$\sum_{n = 1}^{\infty} \frac{2^{w (n)}}{n^{2}} = \frac{5}{2}$

Junaid Ayala 2022-02-25

Convergence of the series
$\sum_{n = 1}^{\infty} (\frac{1}{n^{α}}) (\int_{0}^{\frac{π}{4}} \tan^{n} t d t) x^{n}$

Richie Patterson 2022-02-25

Simplifying:
$\sum_{n = 1}^{\infty} \frac{n a^{n}}{(n - m)! (n + z)} {(x - c)}^{n - m}$

Dachpunkt4cr 2022-02-25

Does the following series absolutely converge, conditionally converge or diverge?
$\sum_{n = 1}^{+ \infty} \sin (n^{2}) \sin (\frac{1}{n^{2}})$

leciazyg 2022-02-25

Show that :
$\sum_{m = 1}^{\infty} \sum_{n = 2^{m - 1}}^{2^{m} - 1} \frac{m}{(2 n + 1) (2 n + 2)} = 1 - γ$

Olivia Vasquez 2022-02-25

That is equivalent to
$π = \sqrt{12} \sum_{n \geq 0} \frac{(- 1)^{n}}{3^{n} (2 n + 1)}$

Kris Patton 2022-02-25

Find the value of
$\sum_{n = 0}^{\infty} \frac{1}{9 n^{2} + 9 n + 2}$

Alexandria Dodd 2022-02-25

Proving that:
$\frac{3}{2} \sum_{k = 1}^{\infty} \frac{4}{k^{3} + k^{2}} = π^{2} - 6$

Zaynah Dunn 2022-02-24

Show that
$\sum_{n = 1}^{\infty} \frac{z^{2^{n}}}{1 - z^{2^{n + 1}}}$
is algebraic.

Adrienne Rowe 2022-02-24

How to evaluate the following sum?
$\sum_{i = 1}^{n} [\frac{3 n - i}{2}]$

lovesickgirlsrd5 2022-02-24

How can I prove this result?
$\sum_{k = 0}^{\infty} \frac{2^{k}}{2^{2^{k}} + 1} = 1$

SnuluddidaPalbx4 2022-02-24

Evaluate:
$\sum_{n = 1}^{\infty} \arctan (\frac{1}{8 n^{2}})$

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