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1. (a) Let an → 2. Prove from first principles (i.e. give a direct e-N proof) that a → 4. (b) Let a → 4. Prove that (an) has a subsequence which converges either to 2 or -2. (c) Let a |an| → 2. 4. Prove from first principles (i.e. give a direct e-N proof) that

1. (a) Let an → 2. Prove from first principles (i.e. give a direct e-N proof) that a → 4. (b) Let a → 4. Prove that (an) has a subsequence which converges either to 2 or -2. (c) Let a |an| → 2. 4. Prove from first principles (i.e. give a direct e-N proof) that

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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B Homework Problems 1. (a) Let an → 2. Prove from first principles (i.e. give a direct e-N proof) that a → 4. (b) Let a 4. Prove that (an) has a subsequence which converges either to 2 or -2. (c) Let a 4. Prove from first principles (i.e. give a direct -N proof) that |an| → 2. 2. For each of the following sequences, (i) Find all accumulation points and compute lim sup and lim inf. (ii) Determine whether the sequence has a convergent subsequence. (iii) Write down a convergent subsequence in case there is one. ={(-1)² otherwise 3. Prove that for any bounded sequences (n) and yn we have lim sup(n + yn) ≤ lim sup(x₂) + lim sup (yn). Give an example that shows that equality may not hold. What is the corresponding inequality for lim inf? if n divisible by 3 an = 4. Consider the series an where an = n=1 0 bn = 1 n(n+3) 1 n² 13' Cn = (-1)" + (a) Find a formula for the kth partial sum, sk=an. (Hint: partial fractions!) n=1 (b) Hence show that the series converges, and find its limit.
Transcribed Image Text:B Homework Problems 1. (a) Let an → 2. Prove from first principles (i.e. give a direct e-N proof) that a → 4. (b) Let a 4. Prove that (an) has a subsequence which converges either to 2 or -2. (c) Let a 4. Prove from first principles (i.e. give a direct -N proof) that |an| → 2. 2. For each of the following sequences, (i) Find all accumulation points and compute lim sup and lim inf. (ii) Determine whether the sequence has a convergent subsequence. (iii) Write down a convergent subsequence in case there is one. ={(-1)² otherwise 3. Prove that for any bounded sequences (n) and yn we have lim sup(n + yn) ≤ lim sup(x₂) + lim sup (yn). Give an example that shows that equality may not hold. What is the corresponding inequality for lim inf? if n divisible by 3 an = 4. Consider the series an where an = n=1 0 bn = 1 n(n+3) 1 n² 13' Cn = (-1)" + (a) Find a formula for the kth partial sum, sk=an. (Hint: partial fractions!) n=1 (b) Hence show that the series converges, and find its limit.
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