Detailed Explanations for Calculus 2 Equations

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?
$\sum_{n = 1}^{+ \infty} \sin (n^{2}) \sin (\frac{1}{n^{2}})$

Zaynah Dunn 2022-02-24

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

Keri Molloy 2022-02-24

How to get the sum of the series
$\sum_{n = 1}^{\infty} \frac{1}{{(4 n^{2} - 1)}^{2}}$ ?

cindakayumn 2022-02-24

Prove that:
$lim_{m \to \infty} \sum_{k = 1}^{m} \frac{2^{- k}}{k} = \log 2$

geekotakuglh 2022-02-24

Evaluate the series:
$\sum_{k = 1}^{\infty} \frac{1}{k {(k + 1)}^{2} k!}$

Ingrid Senior 2022-02-24

How to prove:
$\sum_{n = 1}^{\infty} (\frac{1}{4 n + 1} - \frac{1}{4 n}) = \frac{1}{8} (π - 8 + 6 \ln 2)$

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

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