This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Math 25: Advanced Calculus UC Davis, Spring 2011 Math 25 — Solutions to practice problems Question 1 For n = 0 , 1 , 2 , 3 ,... and 0 ≤ k ≤ n define numbers C n k by C n k = n ! k !( n k )! = n ( n 1) ... ( n k + 1) k ! (for k = 0 and k = n we define C n = C n n = 1). (a) Prove that for all n ≥ 1 and 1 ≤ k ≤ n 1, C n k = C n 1 k 1 + C n 1 k Proof. C n 1 k 1 + C n 1 k = ( n 1)( n 2) ... ( n k + 1) ( k 1)! + ( n 1)( n 2) ... ( n k ) k ! = k + ( n k ) k · ( n 1) ... ( n k + 1) ( k 1)! = n k · ( n 1) ... ( n k + 1) ( k 1)! = C n k (b) Prove by induction on n that for all n ≥ 1, n X k =0 C n k = C n + C n 1 + C n 2 + ... + C n n = 2 n Proof. For n = 1 we have C n + C n 1 = 1+1 = 2 1 , so the claim is true. Let n ≥ 1 be given, and assume the claim is true for that value of n . Then, by using the result of (a) above, we see that n +1 X k =0 C n +1 k = 1 + n X k =1 C n +1 k + 1 = 2 + n X k =1 ( C n k 1 + C n k ) = 2 + n X j =0 C n j + n X k =1 C n k = 2 + n 1 X j =0 C n j 1 ! + n 1 X k =0 C n k 1 ! = 2 + (2 n 1) + (2 n 1) = 2 · 2 n = 2 n +1 , 1 where we used the inductive hypothesis in the transition from the second row to the third. This shows that the claim is true for n +1 and completes the induction. Question 2 (a) Let ( a n ) ∞ n =1 , ( b n ) ∞ n =1 be sequences of real numbers. For each of the follow ing identities, explain what assumptions are needed to ensure that the identity is valid: i. lim n →∞ ( a n + b n ) = lim n →∞ a n + lim n →∞ b n ii. lim n →∞ ( a n · b n ) = lim n →∞ a n · lim n →∞ b n iii. lim n →∞ a n b n = lim n →∞ a n lim n →∞ b n Solution. By a theorem we learned, i. and ii. are valid under the assump tion that the sequences ( a n ) ∞ n =1 and ( b n ) ∞ n =1 are convergent. For iii., one needs the additional assumption that the limit of (...
View
Full
Document
This note was uploaded on 10/16/2011 for the course MATH 34 taught by Professor Wiley during the Winter '11 term at UC Merced.
 Winter '11
 Wiley
 Differential Equations, Calculus, Equations

Click to edit the document details