Assignment #11 -Due 3.14.12doc

Assignment #11 -Due 3.14.12doc - this assignment to answer...

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MAT 345 – Assignment #11 Due 3/19/12 Please write up your solutions on a separate sheet of paper. 1. Today in class, we assumed that 1 1 lim = n in our solution to the final problem. We don’t want to just assume this though; we want to prove this! Prove the following theorem: The constant sequence “c” converges to c. That is, if c is a real number and c s n = for all values of n, then c c s n n n = = lim lim Hint: For all 0 ε , show that there will exist an N where < - = - c c s s n for . N n 2. In class and on homework assignments, we have proved the following: 0 1 lim 2 = n n 0 1 lim 3 / 1 = n n Prove the following theorem: The sequence p n 1 converges to 0 for the values of p greater than 0. That is, 0 1 lim = p n n ) 0 ( p Hint: For all 0 , show that there will exist an N where < - 0 1 p n for . N n Use the limit theorems from class as well as the limit theorems from Problem 1 and Problem 2 in

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Unformatted text preview: this assignment to answer the following problems: 3. Exercise 1(c) on page 52 of the text. 4. Exercise 2 on page 52 of the text. 5. Exercise 3 on page 52 of the text. 6. Prove the following theorem: If a sequence n s converges to s, then the sequence n s converges to s . In other words, if s s n n = ∞ → lim , then s s n n = ∞ → lim . Hint : What needs to be proved is that for ∈ there is a number N so that when n > N, < ∈-s s n . Since ), )( ( ) ( s s s s s s n n n +-=-it follows that s s s s s s n n n +-=-. That means s s s s s s s s s n n n n-≤ +-=-....
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This note was uploaded on 03/20/2012 for the course MATH 345 taught by Professor Michael during the Spring '12 term at University of Massachusetts Boston.

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Assignment #11 -Due 3.14.12doc - this assignment to answer...

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