How can we converge to transcendental numbers beyond e?

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Discussion Overview

The discussion revolves around the convergence to transcendental numbers beyond e, exploring the methods and series that can be used to approach various transcendental numbers. Participants examine the properties of specific numbers like e and π, and the nature of series that converge to these values.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant notes that e is the limit of an exponential function as its argument approaches one, and questions how to find convergence methods for other transcendental numbers.
  • Another participant explains that if a function f(x) converges to F, then e raised to that function converges to e^F, suggesting a broader principle applicable to continuous functions.
  • A different participant mentions that every real number, including transcendental numbers, can be expressed as an infinite decimal expansion, providing an example with π.
  • One participant expresses curiosity about the distinct properties of e and π, highlighting that they have different series that converge to their values, and questions whether it is true that different numbers can have vastly different series even if they are numerically close.
  • Another participant asserts that there are infinite series representations for both e and π, and that different series can converge to the same limit without requiring different limits.
  • A later reply provides examples of series that converge to the same value, illustrating the concept further.

Areas of Agreement / Disagreement

Participants express various viewpoints on the nature of convergence to transcendental numbers, with some agreeing on the existence of multiple series for specific numbers, while the discussion remains unresolved regarding the methods for converging to arbitrary transcendental numbers.

Contextual Notes

The discussion includes assumptions about the properties of functions and series without fully resolving the mathematical steps involved in the convergence processes.

Bob3141592
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e is the limit of an exponential of a number that is approaching one. The exponential makes it want to blow up, but the closeness to 1 keeps that in check. It's really a remarkable number!

My question is, how easy is it to find ways to converge to arbitrary numbers other then e? Almost every number is transcendental so they require such a convergence. A very few of them have fascinatingly concise series, and a remarkable variety of ways to get to that number. But for the rest, how do we get to them?
 
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If f(x) converges to F, then ##e^{f(x)}## converges to ##G=e^F##. For every positive G there is a suitable F=log(G).
This is not limited to exponentiation, this rule works for every continuous function. If f(x) converges to F, then ##2*f(x)## converges to ##G=2*F##.

What do you mean with "get" a number? Here is a transcendal number: ##\pi##.
 
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Every real number can be written as a decimal and, for any irrational number (and most rational numbers), that decimal expansion is an infinite decimal expansion. That is, every such number, which includes all transcendental numbers, can be written as the infinite sequence of it decimal expansions. For example, \pi is the limit of the infinite sequence 3, 3.1, 3.14, 3.141, 3.1415, 3.14159, 3.141592, 3.1415926, ...
 
I have a hard time phrasing this question, so my apologies in advance. e and pi are both very special numbers, with very special properties. Not only do they have really cool ways to define them, but a variety of series converge to exactly their values. That is, a variety of series without arbitrary constants. To me, that's awesomely amazing.

The series that let us know the value of e look nothing like the series that let us know the value of pi. This is true in general, isn't it? Two different numbers could be close to each other in value but have wildly different series convergent to them, and this remains true even as they get increasingly close to each other, as long as they remain different. Is that a true statement?
 
Bob3141592 said:
but a variety of series converge to exactly their values

This is true for many series. For example:

\frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + ... = 2

\frac{1}{1} + \frac{1}{3} + \frac{1}{6} + \frac{1}{10} + \frac{1}{15} + ... = 2

\frac{3}{1} - \frac{3}{2} + \frac{3}{4} - \frac{3}{8} + \frac{3}{16} - ... = 2
 

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