Calculus can be used to approximate the value of pi using infinite series.
One such series, provided in the reference, allows us to calculate approximations of pi:
pi = 2 sqrt(3) (1 - 1/3 3 + 1/5 32 - 1/7 33 + 1/9 34 - ...)
This series involves an alternating sum of terms, where the denominators are odd numbers and the powers of 3 increase with each term. Let's examine how successive terms of this series converge towards pi.
Approximations using the First Five Terms
The reference gives the approximations for pi using the first five terms of the series:
| Number of Terms | Approximation of Pi |
|---|---|
| 1 | 3.46410 |
| 2 | 3.07920 |
| 3 | 3.15618 |
| 4 | 3.13785 |
| 5 | 3.14260 |
As you can see, the approximations get closer to the actual value of pi (approximately 3.14159) as more terms are included. The reference notes that the fifth term approximation (3.14260) differs from pi by only a little more than 0.001.
How Calculus is Involved
While the reference provides the series itself, calculus is crucial in deriving such series for approximating pi. Here's a more general conceptual outline of how calculus plays a role, although the reference doesn't directly provide this derivation:
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Inverse Trigonometric Functions: Functions like arctangent (arctan(x) or tan-1(x)) have well-known series representations (Taylor or Maclaurin series) that can be derived using calculus (specifically, derivatives and integration). The arctangent function is particularly useful because arctan(1) = pi/4.
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Taylor/Maclaurin Series: These series provide polynomial approximations of functions. For example, the Maclaurin series for arctan(x) is:
arctan(x) = x - x3/3 + x5/5 - x7/7 + ...
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Substituting and Solving: Since arctan(1) = pi/4, you might think substituting x = 1 into the arctangent series would give a series for pi directly. However, this series converges very slowly. Therefore, mathematicians often use identities involving arctangent and strategically choose values of 'x' that lead to more rapidly converging series for pi. The provided formula in the reference is likely derived from a more sophisticated use of arctangent identities combined with the Maclaurin series.
Example:
While the reference series doesn't directly come from arctan(x) evaluated at x=1, consider what would happen:
pi/4 = arctan(1) = 1 - 1/3 + 1/5 - 1/7 + 1/9 - ...
pi = 4 * (1 - 1/3 + 1/5 - 1/7 + 1/9 - ...) This is the Leibniz formula for pi. Although it expresses pi, its convergence is too slow to be practical for efficient calculation. The reference formula converges faster.
In summary, calculus provides the tools (derivatives, integrals, series) to develop infinite series representations of functions related to pi (like arctangent), which can then be used to approximate its value to varying degrees of accuracy. The specific formula in the reference is a result of this type of analysis, leading to a more efficient approximation than simpler series like the Leibniz formula.
