Series for coth(x/2) via Bernoulli numbers

In summary, the conversation is about the use of the "Guide to Essential Math" by S.M. Blinder for basic mathematics. The focus is on the section about Bernoulli Numbers and the confusion surrounding equation 7.61. The speaker questions if the transition to this equation is incorrect and asks for clarification on the "explicit" series. They also suggest correcting the equation themselves.
  • #1
Oppie
7
1
Hello,

I've been using "Guide to Essential Math" by S.M. Blinder from time to time to stay on top of my basic mathematics. I'm currently on the section on Bernoulli Numbers. In that section he has the following (snippet below).

Is the transition to equation 7.61 just wrong? The equation just doesn't make sense. If it is wrong, can anyone "see" what he may have had in mind? What is the "explicit" series equal to if not coth(x/2)? Thank you.

1623621585705.png



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  • #2
[tex]\coth x =\frac{1}{x}+\frac{x}{3}+...[/tex]
So why don't you correct (7.61) by yourself ?
 
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  • #3
Oppie said:
What is the "explicit" series equal to if not coth(x/2)?
It's the series for ##\tanh(x/2)##.
 
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1. What is the formula for the series of coth(x/2) via Bernoulli numbers?

The formula for the series is coth(x/2) = 1 + 2/3x + 4/45x^3 + 8/945x^5 + ... + 2^(2n-1)B2n/(2n)!(x/2)^(2n-1), where B2n represents the 2n-th Bernoulli number.

2. What are Bernoulli numbers and how are they related to the series for coth(x/2)?

Bernoulli numbers are a sequence of rational numbers that arise in various mathematical contexts. They are closely related to the series for coth(x/2) because they appear as coefficients in the terms of the series.

3. How do you calculate the Bernoulli numbers needed for the series of coth(x/2)?

The Bernoulli numbers can be calculated using various methods, such as the Euler-Maclaurin formula or the generating function for Bernoulli numbers. Alternatively, you can use a table of known values for the Bernoulli numbers.

4. What is the convergence of the series for coth(x/2) via Bernoulli numbers?

The series for coth(x/2) via Bernoulli numbers converges for all real values of x, as long as x is not a multiple of π. This means that the series is convergent for most values of x, but diverges for a few specific values.

5. Are there any practical applications for the series for coth(x/2) via Bernoulli numbers?

Yes, the series for coth(x/2) via Bernoulli numbers has applications in fields such as physics, engineering, and statistics. It can be used to model various physical phenomena and to solve differential equations in these fields.

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