Question

True or False

a) Every convergent sequence is bounded
b) If lim n→∞ an=0, then the series ∑n=∞ 1 an converges

Answer 1

(a) true - how could it not be so?

(b) again, consider the Harmonic Series

Answer 2

[AI-generated answer — Explain Bot]

a) True. To prove this statement, we can use the definition of a convergent sequence. A sequence {an} is said to converge to a limit L if for any ε > 0, there exists a positive integer N such that for all n > N, |an - L| < ε. Let's assume that the sequence {an} converges to L. This means that for any ε > 0, there exists a positive integer N such that for all n > N, |an - L| < ε. Now, let's consider the absolute value of each term in the sequence. We have |an - L| = max(|an - L|, ε), where ε > 0. This means that for all n > N, |an - L| < max(|an - L|, ε). If we choose ε = 1, we have |an - L| < max(|an - L|, 1). From this inequality, it follows that the sequence {an} is bounded, as there exists a positive number M = max(|a1 - L|, |a2 - L|, ..., |aN - L|, 1) such that for all n in the sequence, |an - L| ≤ M.

b) We need more information to determine if the statement is true or false. The fact that lim n→∞ an = 0 only tells us about the behavior of individual terms in the sequence, but it does not provide sufficient information to determine the convergence or divergence of the series ∑n=∞ 1/an. To determine convergence or divergence, we need to analyze the terms of the series themselves, rather than just their limits.