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I don't think my knowledge of Pi, irrationality, and transcendental numbers in general is complete. I've Googled for a day before posting this question.

Intuitively, I understand why the ratio of the circumference to the diameter is a number less than 4 and how Pi is found by inscribing and circumscribing a regular polygon with a variable number of sides and deriving Pi as a number in between the perimeters of those polygons (and that this window gets smaller after each successive step in an attempt to calculate Pi).

But all intuitive explanations I've encountered sound suspiciously like taking a limit of something.

For example: taking a limit as the number of the sides of a polygon approach infinity (so as it more accurately hugs the surface of a circle), treating that as the circumference and dividing it by the length of the diameter.

But I also know that pi is not algebraic, so I'm not sure if I should be looking for an algebraic expression the limit of which w.r.t. a variable approaches a transcendental number.

So my question, for now, is: Does such a limit exist?

Fredrik Meyer
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Vahram
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    http://mathworld.wolfram.com/PiFormulas.html – Max Oct 19 '15 at 23:59
  • also take this: http://math.stackexchange.com/questions/14113/series-that-converge-to-pi-quickly – Max Oct 20 '15 at 00:01
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    I don't really understand this question. Do you mean $\lim_{x\rightarrow x_0} f(x) = \pi$ for some $f$ and some $x_0$? In that case, $f(x) = \pi$ is a simple example. Do you mean a sequence of numbers that converge to $\pi$? In that case you gave an great answer yourself with the perimeter of polygons. – Brick Oct 20 '15 at 00:22
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    It's a hard question to phrase. I'm pretty sure the OP is struggling with how algebraic expressions/functions can be used to compute a transcendental number; how the limiting behavior of algebraic stuff (like I said, hard to phrase) can get us to transcendental stuff. – pjs36 Oct 20 '15 at 00:31
  • @pjs36, Yes this is what I meant. Sequences are functions of their indices, and so far all the sequences I saw that converge to $\pi$ appear to have their terms related by a function that is algebraic. So how can an algebraic expression converge to a transcendental number? Moreover, as I mentioned, if we take a regular polygon and write down a function describing its perimeter letting the number of its sides approach infinity, we'll get the circumference of a circle, which contains $\pi$, which is a transcendental number. – Vahram Oct 20 '15 at 01:25
  • You can calculate approximate value of pi from Sterling formula also:https://en.wikipedia.org/wiki/Stirling%27s_approximation – NoChance Oct 20 '15 at 08:12

2 Answers2

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You can use the Madhava-Leibniz series:

$$\lim_{k \to\infty}4\sum_{n=1}^k \frac{(-1)^{n+1}}{2n+1}=\pi$$

Alexandre Huat
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Such a function certainly exists. There are many sequences that have a limit of $\pi$.

For example, $3, 3.1, 3.14, 3.141, 3.1415, 3.1415, 3.14159, 3.141592,\dots$ and so on (i.e., following the decimal expansion of $\pi$) has a limit of $\pi$.

There are also more fancy sequences that have a limit of $\pi$. For example, Leibniz figured out one such formula:

https://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80

5xum
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