![]() ![]() This is a pretty obvious statement, so we could call this the california roll theorem. TheoremĪ monotonic bounded sequence must converge. These are some of the ideas that spice up our sushi roll sequences. There are a couple of theorems connecting the ideas of boundedness and convergence for sequences. If a sequence is missing one or both of these bounds, then it's unbounded. ![]() For our sushi sequence, if it is bounded, we can make a bento box with it. If a sequence is bounded above and below, we say it's bounded. We say a sequence is bounded below if there's a value K such that all terms of the sequence are at least K. Just like a function, we say a sequence is bounded above if all terms of the sequence are less than or equal to some value M. That's why, for sequences, we use "increasing" as an abbreviation for "strictly increasing". They are about as interesting as watching water evaporate off a hot road surface in the middle of summer. We don't usually care about nondecreasing sequences. This sequence is neither increasing nor decreasing.īe Careful: Using the word "increasing" to refer to a function is ambiguous because it could mean either nondecreasing or strictly increasing. The terms of the sequence a n = (-1) n bounce back and forth between 1 and -1. It's possible for a sequence to be neither increasing nor decreasing. The sequence is decreasing because the terms get smaller as n gets larger.īe Careful: increasing and decreasing aren't opposites. The sequence a n = n is increasing because the terms get larger as n gets larger. We'll have a monotonically increasing California roll, extra wasabi on the side. If a sequence is increasing or decreasing we call it monotonic because the terms are going only one way. This is like a lop-sided sushi roll where the piece on the right is bigger than the one to its left. We say a sequence is increasing if the terms get larger as n gets larger. These meals are going to look similar to functions, so you may want to review it to see the similarities. Let's look at some of the ways the martial arts master can serve up a sequence to us. Because the sequence is just a coarse-chopped list of numbers made from a function, the sequence acts in ways similar to functions. Those discrete values would form a sequence. Imagine a Kung Fu black belt took a function and chopped through it, leaving only discrete values. For the sake of intuition, it may be helpful to graph the sequence. To determine if a sequence converges or diverges, see if the limitĮxists and is finite. ![]() You'll get up, try to wrangle it, and it'll just throw you over its head again. This is the bull that catches you and throws you over it's head using its horns. The sequence a n = (-1) n diverges because it's indecisive and can't make up its mind whether to be + 1 or -1. This is the bull that gets away from you before you can lasso it. Since the terms aren't getting closer to anything finite, we say the sequence diverges. The sequence a n = n diverges because as n approaches ∞ the terms a n approach ∞ also. Sequences can diverge for different reasons. ![]() Just like a function, if a sequence doesn't converge, we say it diverges. As n gets bigger and we move to the right on the graph, the dots get closer and closer to height 1. The dots will get closer and closer to height L as n gets bigger. If we look at a convergent sequence on a 2-D graph, it looks like a function with a horizontal asymptote. The dots are trying to get to 0 on the number line. If we look at a convergent sequence on a number line, it looks like the dots are getting closer and closer to value L. This is just like convergence for functions. In symbols, a sequence converges to L if. When the terms of a sequence approach some finite value L as n gets bigger, we say the sequence converges to L, as is lasso. The terms of the sequence approach 0 as n approaches ∞. In some of the sequences we graphed, it looked like as n got bigger the values a n approached some particular value. Let's make sure we're comfortable with limits, and let's see which sequences we can stop. Sequences are like bulls at a rodeo waiting to be lassoed, but the divergent ones can't be caught. ![]()
0 Comments
Leave a Reply. |