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**Unformatted text preview: **MATH0201 BASIC CALCULUS MATH0201 BASIC CALCULUS Sequences Dr. WONG Chi Wing Department of Mathematics, HKU MATH0201 BASIC CALCULUS Existence of Irrational Number The Circular Constant Pi Archimedes’ Approximation of Pi Monotone Convergence Theorem The Number e Binomial Theorem A Defining Sequence of e Real Exponents and related Functions Power Functions Exponential Functions Logarithmic Functions Reference § 4.1–2 of the textbook. MATH0201 BASIC CALCULUS Existence of Irrational Number “All is Number” Pythagorean School (500 B.C.) I Number means Positive Integer. I The nowadays rational numbers were thought as ratio. I Any two line segments are commensurable. However, edge and diagonal of a square is incommensurable! MATH0201 BASIC CALCULUS Existence of Irrational Number Theorem 1 p 2 is a irrational number. Proof. (Optional) Suppose that p 2 = p = q , where p and q are relative prime positive integers. It follows that p is even. It turns out that q is even as well. Contradiction! MATH0201 BASIC CALCULUS Existence of Irrational Number Babylonian Approximation Method p 2 can be approximated by iterating r n = 1 2 r n 1 + 2 r n 1 ; r 1 = 1 : n r n 2 = r n 1 1 2 2 3 = 2 4 = 3 3 17 = 12 1 : 41667 24 = 17 4 577 = 408 1 : 41422 816/577 MATH0201 BASIC CALCULUS The Circular Constant Pi The ancients has observed that if r i , C i , and A i ( i = 1 ; 2) are the radii, the circumferences, and the areas of two circles, then C 1 2 r 1 = C 2 2 r 2 = 1 and A 1 r 2 1 = A 2 r 2 2 = 2 : Archimedes may be the first one who gave a rigorous proof that 1 = 2 =: . Theorem 2 A real number x is rational if and only if its decimal expression is either terminating or recurring. MATH0201 BASIC CALCULUS The Circular Constant Pi The (approximated) value of has been searching for more than 2000 years. Theorem 3 (Lambert, 1761) is a irrational number MATH0201 BASIC CALCULUS The Circular Constant Pi Archimedes’ Approximation of Pi Consider a circle with radius 1 2 , of which the circumference is . The perimeter of a regular n –gon inscribed in that circle is P n = n sin 180 n : n P n 6 3 : 0000 12 3 : 1058 24 3 : 1326 48 3 : 1394 96 3 : 1410 192 3 : 1415 384 3 : 1416 MATH0201 BASIC CALCULUS The Circular Constant Pi Monotone Convergence Theorem Plot the values of P n against n Observe that I P n is increasing. I P n is bounded above. I P n can be made as close to as we please by increasing n . MATH0201 BASIC CALCULUS The Circular Constant Pi Monotone Convergence Theorem Theorem 4 (Monotone Convergence Theorem) If a sequence f a n g is increasing and is bounded above (by a number M) a 1 < a 2 < a 3 < < M then it has a limit ‘ lim n !1 a n = ‘: That is, a n can be made as close to ‘ as we please by increasing n....

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