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Consider the number C(n,r) given by n! C(n, r) = (2.1) (n – r) ! -r ! where n and r are integers satisfying n 2 2 and n 2 r 2 0 (0! = 1). In particular, C(n, 0) = C(n, n) = 1. ( 2.2) [ You are not required to show ( 2.2).] Let P, := II C(n, k) = C(n, 0)xC(n, 1) × ·… x C(n, n) k- 1 ... for n = 2, 3, .... Does the following limit { Pn + 1 × P, lim exist ? (2.3) You are not required to prove your answer Moreover, if your answer is “Yes", then find the limit , and write down your answer in terms of some of the fundamental numbers like 7, e (base for the natural log . ), 1, 2, .. (note that not all of these number may be present ). Here you are required to justify your answer (on the value of the limit ) . You may take the following limit ) - lim 1 + = e (2.4) for granted [ that is, you are not required to show (2.4)].

Consider the number C(n,r) given by n! C(n, r) = (2.1) (n – r) ! -r ! where n and r are integers satisfying n 2 2 and n 2 r 2 0 (0! = 1). In particular, C(n, 0) = C(n, n) = 1. ( 2.2) [ You are not required to show ( 2.2).] Let P, := II C(n, k) = C(n, 0)xC(n, 1) × ·… x C(n, n) k- 1 ... for n = 2, 3, .... Does the following limit { Pn + 1 × P, lim exist ? (2.3) You are not required to prove your answer Moreover, if your answer is “Yes", then find the limit , and write down your answer in terms of some of the fundamental numbers like 7, e (base for the natural log . ), 1, 2, .. (note that not all of these number may be present ). Here you are required to justify your answer (on the value of the limit ) . You may take the following limit ) - lim 1 + = e (2.4) for granted [ that is, you are not required to show (2.4)].

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.4: Mathematical Induction
Problem 29E
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Question 2 Consider the number C(n, r) given by n! C(n,r) (2.1) (n – r) !·r ! where n and r are integers satisfying n 2 2 and n 2r 2 0 ( 0! = 1). In particular, С (п, 0) [ You are not required to show (2.2).] Let С (п, п) 3D 1. (2.2) P, := П С(п, К) C (n, 0) x C (n, 1)× ... x C (n, n) k = 1 for n 2, 3, Does the following limit ..... { Pn + 1 x Pn - 1 [Pn ]? lim n + 00 exist ? ( 2.3) You are not required to prove your answer Moreover, if your answer is “Yes", then find the limit, and write down your answer in terms of some of the fundamental numbers like 7, e (base for the natural log .), 1, 2, ... (note that not all of these number may be present ). Here you are required to justify your answer (on the value of the limit ). You may take the following limit lim (1 + ) - = e (2.4) for granted [ that is, you are not required to show (2.4)].
Transcribed Image Text:Question 2 Consider the number C(n, r) given by n! C(n,r) (2.1) (n – r) !·r ! where n and r are integers satisfying n 2 2 and n 2r 2 0 ( 0! = 1). In particular, С (п, 0) [ You are not required to show (2.2).] Let С (п, п) 3D 1. (2.2) P, := П С(п, К) C (n, 0) x C (n, 1)× ... x C (n, n) k = 1 for n 2, 3, Does the following limit ..... { Pn + 1 x Pn - 1 [Pn ]? lim n + 00 exist ? ( 2.3) You are not required to prove your answer Moreover, if your answer is “Yes", then find the limit, and write down your answer in terms of some of the fundamental numbers like 7, e (base for the natural log .), 1, 2, ... (note that not all of these number may be present ). Here you are required to justify your answer (on the value of the limit ). You may take the following limit lim (1 + ) - = e (2.4) for granted [ that is, you are not required to show (2.4)].
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