Circle Series Reciprocal: Taking the Reciprocal of an Infinite Series

In summary, UnitSuppose f(x) is analytic in a domain containing x_{0} and f(x_{0}) \neq 0. Then, 1/f(x) is also analytic. Both of these functions have Taylor series expansions around x_{0}:f(x) = \sum_{n = 0}^{\infty}{a_{n} \, (x - x_{0})^{n}}, \; a_{0} \neq 0\frac{1}{f(x)} = \sum_{n = 0}^{\infty}{b_{n} \, (x - x_{0})^{n}}Multiplying
  • #1
Unit
182
0
I understand how this works:

[tex]\cos x = \frac{1}{0!} - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \frac{x^8}{8!} - \frac{x^{10}}{10!} + \ldots[/tex]

But what about this?

[tex]\frac{1}{\cos x} = \frac{1}{0!} + \frac{x^2}{2!} + \frac{5x^4}{4!} + \frac{61x^6}{6!} + \frac{1385x^8}{8!} + \frac{50521x^{10}}{10!} + \ldots[/tex]

Is there a way to take the reciprocal of an infinite series or is it necessary to take subsequent derivatives of secant and write the Taylor expansion that way?

Thanks,
Unit
 
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  • #2
Suppose [itex]f(x)[/itex] is analytic in a domain containing [itex]x_{0}[/itex] and [itex]f(x_{0}) \neq 0[/itex]. Then, [itex]1/f(x)[/itex] is also analytic. Both of these functions have Taylor series expansions around [itex]x_{0}[/itex]:

[tex]
f(x) = \sum_{n = 0}^{\infty}{a_{n} \, (x - x_{0})^{n}}, \; a_{0} \neq 0
[/tex][tex]
\frac{1}{f(x)} = \sum_{n = 0}^{\infty}{b_{n} \, (x - x_{0})^{n}}
[/tex]

Multiplying the two series term by term, we get:

[tex]
\sum_{n = 0}^{\infty}{\left(\sum_{m = 0}^{n}{a_{n - m} \, b_{m}}\right) \, (x - x_{0})^{n}} = 1
[/tex]

from where, we get the following conditions for the coefficients [itex[\{b_{n}\}[/itex]:

[tex]
a_{0} \, b_{0} = 1 \; \Rightarrow \; b_{0} = \frac{1}{a_{0}}
[/tex]

[tex]
\sum_{m = 0}^{n}{a_{n - m} \, b_{m}} = 0, \; n \ge 1 \; \Rightarrow \; \sum_{m = 1}^{n}{a_{n - m} \, b_{m}} = -\frac{a_{n}}{a_{0}}
[/tex]

The matrix of this system is:

[tex]
A = \left(\begin{array}{ccccc}
a_{0} & 0 & \ldots & 0 & \ldots \\

a_{1} & a_{0} & \ldots & 0 & \ldots \\

\ldots & & & & \\

a_{n - 1} & a_{n - 2} & \ldots & a_{0} & \ldots \\

\ldots & & & &
\end{array}\right)
[/tex]

It is an infinitely dimensional lower triangular matrix. If we restrict it to the first n rows, then the determinant is simply [itex]\det A_{n} = a^{n}_{0} \neq 0[/itex], so the matrix is never singluar and the inverse always exists. However, it is pretty difficult to find an analytical form for it in the general case.
 
  • #3
Dickfore is correct about the difficulties in finding inverse of infinity series. I have written a paper about finding power series of tan x + sec x. And I think Unit may take a look in it. For the secant series, it just corresponds to the even power of the series, as tan x is odd, and sec x is even.

http://www.voofie.com/content/117/an-explicit-formula-for-the-euler-zigzag-numbers-updown-numbers-from-power-series/" [Broken]
 
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1. What is a circle series reciprocal?

A circle series reciprocal is a mathematical concept where the reciprocal (1/x) of an infinite series is taken and the resulting series is summed. This is also known as the reciprocal of an infinite series.

2. How is the reciprocal of an infinite series calculated?

The reciprocal of an infinite series is calculated by taking the reciprocal of each term in the series and then summing the resulting series. For example, if the infinite series is 1 + 1/2 + 1/3 + 1/4 + ..., the reciprocal series would be 1/1 + 1/2 + 1/3 + 1/4 + ... = 1 + 1/2 + 1/3 + 1/4 + ...

3. What is the purpose of taking the reciprocal of an infinite series?

The purpose of taking the reciprocal of an infinite series is to study the behavior of the series and determine if it converges or diverges. It can also be used to find the value of the original series if it is known that the reciprocal series converges.

4. Is the reciprocal of an infinite series always convergent?

No, the reciprocal of an infinite series is not always convergent. It depends on the original series and its behavior. If the original series is convergent, then the reciprocal series will also be convergent. However, if the original series is divergent, the reciprocal series may or may not be convergent.

5. Can the reciprocal of an infinite series be used in real-world applications?

Yes, the reciprocal of an infinite series has many real-world applications, such as in electrical engineering, physics, and economics. It can be used to model natural phenomena, analyze data, and make predictions. It is also used in various mathematical proofs and calculations.

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