(^) k Recall that for numbers 0 ≤ k ≤ n the binomial coefficient (^) is defined as n! k! (n−k)! Question 1. (1) Prove the following identity: (22) + (1121) = (n+1). (2) Use the identity above to prove the binomial theorem by induction. That is, prove that for any a, b = R, n (a + b)" = Σ (^) an- n-kyk. k=0 n Recall that Σ0 x is short hand notation for the expression x0+x1+ +xn- (3) Fix x = R, x > 0. Prove Bernoulli's inequality: (1+x)" ≥1+nx, by using the binomial theorem. - Question 2. Prove that ||x| - |y|| ≤ |x − y| for any real numbers x, y. Question 3. Assume (In) nEN is a sequence which is unbounded above. That is, the set {xn|nЄN} is unbounded above. Prove that there are natural numbers N] < N2 < N3 < ... so that for all k € N, xnk ≥ k. Question 4. (1) Assume that the sequence (In) n≥1 converges to x. Prove that the sequence (|xn|) n≥1 converges to |x|. (2) Show by means of an example that the converse is not necessarily true. That is, if (n) n≥1 converges to |x|, that need not imply that (xn)n≥1 converges to x. Question 5. Let n₁ < n2 < N3 < nk >k for all k Є N. be natural numbers (nk Є N). Prove that

(^) k Recall that for numbers 0 ≤ k ≤ n the binomial coefficient (^) is defined as n! k! (n−k)! Question 1. (1) Prove the following identity: (22) + (1121) = (n+1). (2) Use the identity above to prove the binomial theorem by induction. That is, prove that for any a, b = R, n (a + b)" = Σ (^) an- n-kyk. k=0 n Recall that Σ0 x is short hand notation for the expression x0+x1+ +xn- (3) Fix x = R, x > 0. Prove Bernoulli's inequality: (1+x)" ≥1+nx, by using the binomial theorem. - Question 2. Prove that ||x| - |y|| ≤ |x − y| for any real numbers x, y. Question 3. Assume (In) nEN is a sequence which is unbounded above. That is, the set {xn|nЄN} is unbounded above. Prove that there are natural numbers N] < N2 < N3 < ... so that for all k € N, xnk ≥ k. Question 4. (1) Assume that the sequence (In) n≥1 converges to x. Prove that the sequence (|xn|) n≥1 converges to |x|. (2) Show by means of an example that the converse is not necessarily true. That is, if (n) n≥1 converges to |x|, that need not imply that (xn)n≥1 converges to x. Question 5. Let n₁ < n2 < N3 < nk >k for all k Є N. be natural numbers (nk Є N). Prove that

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.5: The Binomial Theorem

Problem 26E

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