Proving Monotonicity of Sequences: A Comprehensive Guide

Understanding the behavior of sequences is a fundamental concept in mathematics, especially in calculus and analysis. Two key types of sequences that mathematicians often study are monotone increasing and monotone decreasing sequences. This article aims to guide you through the process of proving whether a specific sequence is monotonic, with a focus on the sequence defined as an 1 - 1/10n.

Introduction to Monotonic Sequences

A sequence {an} is said to be monotonic increasing if an 1 ≥ an for all terms in the sequence. Conversely, a sequence is monotonic decreasing if an 1 ≤ an for all terms. In this article, we will explore how to prove the monotonicity of the specific sequence defined as an 1 - 1/10n.

Definition of the Sequence

The sequence in question is defined by the formula:

an 1 - 1/10n

where n 1, 2, 3, ...

Analysis of the Sequence

To determine whether the sequence is monotonic decreasing or increasing, we need to compare consecutive terms. We start by considering the difference between two consecutive terms an 1 and an.

an 1 - an left(1 - 1/10n 1right) - left(1 - 1/10nright)

Simplifying the expression:

an 1 - an -1/10n 1 1/10n

an 1 - an 1/10n - 1/10n 1

an 1 - an 1/10n(1 - 1/10)

an 1 - an 1/10n(9/10)

an 1 - an (1/10n) * (9/10)

Since (1/10n) * (9/10) > 0 for all n ≥ 1, we conclude that:

an 1 - an > 0

an 1 > an

Conclusion

The sequence an 1 - 1/10n is monotonically increasing as each term is greater than the previous term for all n ≥ 1.

Additional Insights

There might be some confusion when comparing this sequence to another sequence defined by 0.6, 0.66, 0.666, ... which can be represented as xn 0.6 0.06 0.006 ... 0.6 * 10-n. This sequence is also monotonic increasing as any term minus the previous term is positive.

For a sequence {sn}, to determine if it is monotonically increasing or decreasing, you can compute the ratio sn 1/sn. If sn 1/sn ≥ 1, the sequence is decreasing; if sn 1/sn ≤ 1, the sequence is increasing. For the sequence sn 2/3(1 - 10-n), the sequence is monotonically increasing as shown below:

sn 1/sn (2/3)(1 - 10-n-1) / (2/3)(1 - 10-n) (1 - 10-n-1) / (1 - 10-n) (1 - 10-n-1) / (1010-n - 1) 10n(1 - 10-n-1) / (10n(10 - 10-n-1)) (10n - 10-n-1) / (10n - 10-n-1) 10n/10n-1 10

Since the ratio is greater than 1, the sequence is monotonically increasing.