Use an+1-an to show that the given sequence {an} is strictly increasing or strictly decreasing.

Use an+1-an to show that the given sequence {an} is strictly...

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Key Concepts

Sequence

A sequence is an ordered list of elements, typically numbers, where each element is associated with a natural number index. Understanding sequences is critical because they serve as the foundation for studying limits, convergence, and the behavior of functions in calculus and analysis.

Monotonicity

Monotonicity refers to the property of a sequence where its terms consistently increase or decrease. A sequence is strictly increasing if every term is less than the following term, and strictly decreasing if every term is greater than the subsequent one. This concept is central to analyzing the convergence and stability of sequences.

Difference of Consecutive Terms

Analyzing the difference between consecutive terms, expressed as a??? - a?, is a common method to determine the monotonic behavior of a sequence. If this difference is positive for all n, the sequence is strictly increasing; if it is negative, the sequence is strictly decreasing.

Algebraic Simplification

Algebraic simplification involves rearranging and reducing expressions to reveal underlying properties such as sign and magnitude. In the context of sequences, simplifying the difference between consecutive terms helps in identifying whether the sequence is increasing or decreasing by clearly showing the sign of the difference for all terms.

Transcript

00:01 Suppose i have a sequence defined by n over 2n plus 1, where the n values start at 1 and go to positive infinity.

00:06 And i'm trying to figure out if this sequence is monotonic.

00:09 What does monotonic mean? it means as the end values increase, as i plug in higher and higher in values into the ordered sequence, what i could do is the numbers are strictly increasing, strictly decreasing, or maybe non -increasing, which means they increase, but maybe some numbers are equal.

00:27 For example, 1, 2, 2, 3, 3, 4, and so on.

00:30 And so forth.

00:32 So one trick that we learned to see if they're increasing and decreasing is analyze the relationship between a, n plus 1, and the nth term.

00:40 So n plus first term, replace n with n plus 1 over 2 times n plus 1 plus 1 minus the nth term, n over 2 n plus 1.

00:52 Okay.

00:53 So what i can do here, let's just simplify a little bit.

00:55 This is n plus 1 over if i distribute 2n plus 2 or 2 and plus 3.

01:01 Minus n over n plus 1.

01:04 When i'm adding in subtracting fractions, i can have the same denominator.

01:09 So i'm going to multiply the opposite denominator.

01:11 And if i multiply it to the top and bottom of one fraction, it's the same as multiplying by one to each fraction.

01:19 So i can factor foil here...

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