Is there a function whose limit approaches Pi?

Question

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?

Answer 1

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

Answer 2

You can use the Madhava-Leibniz series:

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