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The Bartle-Sherbert bounds 2.828 < pi < 3.185

Example

Let $\gamma=\pi/2$ . Then $\sqrt2<\gamma<\sqrt{6-2\sqrt3},\qquad 2.828<\pi<3.185.$

Facts & Assumptions

Given: The first positive cosine zero $\gamma=\pi/2$ .

[L1]

$\cos$ is strictly decreasing on $[0,2]$ , with unique zero $\gamma$ (Pi as twice the smallest positive zero of cosine, Sine is positive and cosine is strictly decreasing on (0,2), with cos 2 at most -1/3).

[L2]

The cosine power series and alternating-series remainder bounds are those of Sine and cosine defined by their real power series and The alternating series test: if $(b_k)$ is nonincreasing with $b_k \to 0$ then $\sum_{k} (-1)^{k} b_k$ converges, the sum lies between any two consecutive partial sums, and the error after $n$ terms is at most $b_n$ .

[L3]

Nonnegative square roots exist and preserve comparisons after squaring (Square roots exist: a unique $\sqrt{a} \ge 0$ with $(\sqrt{a})^2 = a$ ; the positives are $\{x^2 : x \neq 0\}$ ).

Verification

technique · direct

1.1

At $x=\sqrt2$ , the first two cosine terms cancel and the alternating tail beginning with $x^4/4!$ is positive, so $\cos(\sqrt2)>0$ .

L2L3

1.2

Put $a=\sqrt{6-2\sqrt3}$ . Then $1-a^2/2+a^4/24=0$ , and the remaining alternating cosine tail begins negative with decreasing absolute terms, so $\cos a<0$ .

L2L3algebra

2.1

Strict decrease and [L1] give $\sqrt2<\gamma<a$ .

step 1.1step 1.2L1

3.1

Since $(707/500)^2<2$ , $2\sqrt2>707/250$ . Since $(433/250)^2<3$ and $(637/400)^2>6-2(433/250)$ , one has $a<637/400$ , hence $2a<637/200$ .

step 2.1L3algebra

4.1

Doubling the bounds of step 2.1 proves the displayed decimal bounds for $\pi$ .

step 2.1step 3.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 71 results over 19 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

C. Schmeiser, Introduction to Analysis (standard reference, not scraped)
R. Bartle and D. Sherbert, Introduction to Real Analysis (standard reference, not scraped)
H. Zeisel, lim sin(x)/x and the definition of pi (standard reference, not scraped)