Does "pi" have a limit like "e"?

Question

limit as "n" approaches infinity; (1+(1/n))^n = "e"

sorry, but even though some limits involve of "pi" have triganometery functions built in don't really count as true limits, becasue "pi" is incorbriated into each function, meaning the end result of each limit would be "pi"="pi"

is there a limit that doesn't evolve "pi"?

Answer 1

You probably meant to ask: Can pi be expressed as a limit?

Yes, it can. And in many different ways, via summations, sequences, and integrations. Here are some of the most popular ones:

summ(n=0,inf) (-1)^n/(2n+1) = 1/1 - 1/3 + 1/5 - 1/7 + 1/9 - ... = pi/4

prod(n=1,inf) [(n+1)/n]^[(-1)^(n-1)] = 2/1 * 2/3 * 4/3 * 4/5 * 6/5 * 6/7 * 8/7 * 8/9 * ... = pi/2

sqrt(2)/2 * (sqrt(2+sqrt(2))/2 * (sqrt(2+sqrt(2+sqrt(2)))/2 * ... = 2/pi

integ(-1,1) sqrt(1-x^2)dx = pi/2

Many more exist using a wide variety of methods and with varying levels of complexity. Some of them are the basis for the algorithms which let computers calculate pi to a large number of digits.

Answer 2

You are using terminology mistkenly: e does not "have" a limit, and neither does pi. What you probably mean is "can pi appear AS the limit of an algebraic calculation?". If this is what you meant, the answer is YES: a simple example (from many, many others) is:

pi = 4 * (1 - 1/3 + 1/5 - 1/7 + 1/9...)

both e and pi are themselves TRANSCENDENTAL numbers. In mathematics, a transcendental number is any number that is not algebraic, that is, not the solution of a non-zero finite polynomial equation with integer coefficients. The most prominent examples of transcendental numbers are pi and e. The algebraic formula I gave above is NOT finite: it has an infinite number of terms, and pi is not its "solution" -- it is the formula's "limit". (That is the difference between a solution and a limit)

Transcendental numbers are always irrationals, but not all irrational numbers are transcendentals: for example, the square root of 2 is irrational, but is a solution of the polynomial x**2 − 2 = 0. Irrational numbers expressed as decimals never cycle (that is, repeat). IF the decimal expansion stops or cycles, the number is a rational, and can be expressed in one instance by a/b where both a and b are integers.

Answer 3

Yes pi, can be expressed as a limit. Take a circle with radius = 1
The area of the circle = pi*r^2 = pi*1^2 =pi

Now inscribe an n-sided regular polygon inside the circle. As n increases to infinity, the area of the polygon will approach the area of the circle, which is =pi

The area of the polygon = n/2*sin(360/n)
n = 4 --->2
n =8---> 2.828427125..
n =96--->3.139350203
n =1000--->3.141571983
n=10000--->3.141592447
n =100000--->3.141592652

Answer 4

yes... everything has a limit... there is a slightly higher number that pi will never reach but get closer to just like e does.... they are irrational numbers but that is besides the point. But if you are looking at the graph of e^x as a limit as x approaches negative infinty as being 0, then we would have to look at pi^x and I am sure you would get a graph that has a similiar form as ALL exponitial graphs go through the point (0,1) and the boring graph is 1^x as it is the graph of y=1....pi^x climbs faster than e^x as pi is a larger number than e...

But the graphs y=e and y=pi are boring straight lines.....

Answer 5

I assume by 'like e' you mean something like these formulas for e:

e = lim (n->oo) (1+1/n)^n

or

e = sum(i=0...oo) 1/i!

If so, there are some limits for pi:

pi = sum (i=0..oo) [4/(8i + 1) - 2/(8i + 4) - 1/(8i + 5) - 1/(8i + 6)]/16^i

Also, the formula for atan(x) is:

atan(x) = x - x^3/3 + x^5/5 - x^7/7 ...

If x = 1, then atan(x)=pi/4, so:

pi = 4*atan(1) = 4 - 4/3 + 4/5 - 4/7 + 4/9 ...

Answer 6

Pi Li Mit

Answer 7

I dont think so. There is an infinite FORMULA to pi, contrary to C/d. I think it doesn't have any limit:

4(1-1/3+1/5-1/7+1/9-1/11...)

Answer 8

No "pi" has no limit. pi is 1.1416. it was run an a computer day any night 7 days a week for two years and stayed constant. 1.1416 is rounded off, but when ran on the computer all number places remained the same.

Answer 9

pi = limit as "n" approaches infinity of the circumference of a regular n-sided polygon divided by the largest distance between two of its vertices.

Answer 10

I have read about the limit of 'e'. But I have not yet come across limit of pi.