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Unformatted text preview: Sequences, Series and Foundations These notes by Mikhail Safonov serve as a supplementary material to the textbook by Weyne Richter “Sequences, Series and Foundations. Math 2283 and 3283W” Chapter 1. Truth, Falsity and Mathematical Induction 1 Truth Tables The mathematical statements P,Q, etc. can be treated as variables which can take one of two values: ”T”=”true” and ”F”=”false”. If we further associate ”T” with ”1”, and ”F” with ”0”, then we have the following simple rules: P & Q = PQ, ¬ P = 1 P, P ∨ Q = ¬ (( ¬ P ) & ( ¬ Q )) = 1 (1 P )(1 Q ) = P + Q PQ. Exercise 1.1 Simplify the statement P & Q = P ∨ Q. Solution. Since P and Q have values 1 and 0 , we always have P = P 2 and Q = Q 2 . Therefore, the given statement can be rewritten as PQ = P 2 + Q 2 PQ, so that ( P Q ) 2 = 0 , and P = Q. 4 Mathematical Induction Definition 4.1 For n ∈ N and k = 0 , 1 ,... ,n, the number of combinations of k objects from a set of n objects is n k ¶ = n ! k !( n k )! , where 0! = 1! = 1 , and n ! = 1 · 2 · ... · n for n ≥ 2 . The following formula follows either by easy direct calculation, or by considering separately subsets of { 1 , 2 ,... ,n,n + 1) which (i) contain n + 1 , and (ii) do not contain n + 1 . Lemma 4.2 For n ∈ N and k = 1 ,... ,n, we have n + 1 k ¶ = n k ¶ + n k 1 ¶ . In turn, using induction and this formula, one can get the following important Newton’s binomial formula . For this reason ( n k ) are also called the binomial coefficients. Theorem 4.3 (Newton’s binomial formula) For any n ∈ N and (real or complex) a and b, we have ( a + b ) n = n X k =0 n k ¶ a n k b k = a n + n 1 ¶ a n 1 b + n 2 ¶ a n 2 b 2 + ··· + n n 1 ¶ ab n 1 + b n . 1 In particular, taking a = b = 1 , we get 2 n = n X k =0 n k ¶ . This equality has a simple interpretation: the number of all subsets of a set of n objects (including the empty set) is 2 n . Therefore, the notation 2 S =”all subsets of a given set S ” makes sense even for infinite sets S. Definition 4.4 The sets S 1 and S 2 are equivalent if there is a onetoone function f : S 1 → S 2 , i.e. (i) from s ,s 00 ∈ S 1 and s 6 = s 00 it follows f ( s ) 6 = f ( s 00 ) ; (ii) for each s 2 ∈ S 2 , there is s 1 ∈ S 1 such that f ( s 1 ) = s 2 . Theorem 4.5 (Cantor) For an arbitrary set S, the sets S and 2 S are not equivalent. Proof. Suppose otherwise. Then there is a onetoone function f : S → 2 S . Denote S = { s ∈ S : s / ∈ f ( s ) } ∈ 2 S . By our assumption, S = f ( s ) for some s ∈ S. Consider two possible cases (i) s ∈ S , and (ii) s / ∈ S . In the case (i), s ∈ f ( s ) , hence by definition of S , we have s / ∈ S = f ( s ) , i.e. case (ii)....
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 Spring '09
 Taylor Series, Mathematical Induction, Limit of a sequence, Sn, Rolle

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