sprstph14 said:Suppose that ak is a decreasing sequence and (ak) approaches 0. Prove that for every k in the natural numbers, ak is greater than or equal to 0.
I was thinking I should assume the sequence is bounded below by 0 and do a proof by contradiction.
Any suggestions?
A sequence in real analysis is a list of numbers that are arranged in a specific order. It is denoted by {an} or (an) and each number in the sequence is represented by the subscript n. The order of the numbers is important and can be infinite.
A convergent sequence in real analysis is a sequence in which the terms get closer and closer to a specific limit as n approaches infinity. This limit is known as the limit of the sequence and is denoted by lim an = L. In other words, the terms of the sequence approach a specific value as n gets larger.
A divergent sequence in real analysis is a sequence in which the terms do not approach a specific limit as n approaches infinity. This means that the terms of the sequence either increase or decrease without approaching a specific value. Divergent sequences can be either unbounded or oscillating.
To test for convergence or divergence of a series in real analysis, you can use various tests such as the ratio test, the root test, or the comparison test. These tests compare the series to a known series with known convergence or divergence properties to determine if the series in question also converges or diverges.
A sequence is a list of numbers arranged in a specific order, while a series is the sum of the terms in a sequence. In other words, a series is the result of adding up all the terms in a sequence. While a sequence can be convergent or divergent, a series is said to be convergent if its corresponding sequence is convergent.