[tex] \frac{ \zeta(2)}{2} - \frac{ \zeta(3)}3 + \frac{ \zeta(4)}{4} - \frac{ \zeta(5)}{5} + \cdots \cdots \\ [/tex]

Answers

Answer 1

Recall the Weierstrass product definition of the gamma function,

[tex]\displaystyle \frac1{\Gamma(z)} = ze^{\gamma z} \prod_{n=1}^\infty \left(1+\frac zn\right) e^{-z/n}[/tex]

from which we can obtain the log-gamma (i.e. logarithm of gamma) function,

[tex]\displaystyle -\ln\Gamma(z) = \ln(z) + \gamma z + \sum_{n=1}^\infty \left(\ln\left(1 + \frac zn\right) - \frac zn\right)[/tex]

where [tex]\gamma[/tex] is the Euler-Mascheroni constant.

Let [tex]z=1[/tex], so that

[tex]\displaystyle \gamma = \sum_{n=1}^\infty \left(\frac1n - \ln\left(1 + \frac 1n\right)\right)[/tex]

Recall the power series

[tex]\displaystyle \ln(1 + x) = - \sum_{n=1}^\infty \frac{(-x)^n}n[/tex]

Along with the definition of the Riemann zeta function,

[tex]\displaystyle \zeta(a) = \sum_{b=1}^\infty \frac1{b^a}[/tex]

we find that

[tex]\displaystyle \gamma = \sum_{n=1}^\infty \left(\frac1n + \sum_{m=1}^\infty \frac1m\left(-\frac1n\right)^m\right) \\\\ ~~~~ = \sum_{n=1}^\infty \sum_{m=2}^\infty \frac1m \left(-\frac1n\right)^m \\\\ ~~~~ = \sum_{m=2}^\infty \frac{(-1)^m}m \sum_{n=1}^\infty \frac1{n^m} \\\\ ~~~~ = \sum_{m=2}^\infty \frac{(-1)^m \zeta(m)}m[/tex]

so the original sum's value is [tex]\boxed{\gamma\approx0.577216}[/tex].

Related Questions

someone please help me out with this need for important grade

Answers

Answer:

Step-by-step explanation:

1605 - 75

1530 / 255 = 6

Can somebody please help me I'm trying to find out what 2 + 2 is

Answers

Answer:

2+2 = 4

Step-by-step explanation:

Let's visualize this using # signs

we have two groups of two # signs, so let's map that out

If you count the individual # signs, it is 4. You can also use this model to figure out 2 *2 (4)

I hope this helps :)

Answer:

Step-by-step explanation:

2+2

2+1+1

3+1

select two points that lie on the same line throught (0,0), then select an equation that describes the relationship between the coordinates x and y

Answers

Two collinear points on a line are given in the table below:

x, y

0 0 (0,0)

2 1 (2,1)

Select the points that lie on the line. and the relationship between the coordinates x and y.

(4,2)

(4, 3)

(7,2)

4/8, 2/8

Answer:

(4,2) and ([tex]\frac{4}{8}[/tex],[tex]\frac{2}{8}[/tex]) are lie on the line, and y = [tex]\frac{1}{2}[/tex] x , is showing the relationship of x and y coordinates of given line.

Given that

The co-ordinates of line is (0,0) and (2,1).

To find

The points that lie on the line, and The relationship between the coordinates x and y.

So, according to the question

We have,

The X and Y co-ordinates of line (0,0) and (2,1).

For finding any point that lie on line, first we have to find the slope of given line,

So, we know that

slope (m) = [tex]\frac{y_2-y_1}{x_2-x_1}[/tex]

Where,

x₁ = 0 and x₂ = 2

y₁ = 0 and y₂ = 1

Now, putting the values, we will get

(m) = [tex]\frac{1-0}{2-0}[/tex] = [tex]\frac{1}{2}[/tex]

m = [tex]\frac{1}{2}[/tex]

We know that, the equation of line

y-y₁ = m(x- x₁ )

y-0 = [tex]\frac{1}{2}[/tex] (x- 0)