The key is that the variable n is tending toward infinity (∞), so most of the same techniques that worked to find horizontal asymptotes also work in this new setting.īy the way, now is a great time to review: How do you Find the Horizontal Asymptotes of a Function? Example There are a few standard tricks to working out these kinds of limits. Convergence and LimitsĪs you can see from the definition, testing the convergence of a sequence requires taking a limit. There’s no limit to the values, quite literally. is divergent because the values simply get larger without bound. If the limit does not exist, then we say that the sequence diverges (or is divergent).įor example, the sequence of natural numbers, 1, 2, 3, 4, 5, …. However to rigorously prove this requires a more careful argument.įor our purposes, we will simply state this limit fact without proof. The higher n is, the closer 1/n will get to 0. How can we establish this fact? Well intuitively speaking, if you plug in a very large value of n into the formula 1/ n, what do you get? A little experimentation may lead you to the guess that 1/ n converges to 0. The harmonic sequence ( a n = 1/ n) converges to 0. Just like limits of functions, we use the “lim” notation. To be more precise, we say that the limit (as n → ∞) of the convergent sequence exists (and equals a). We say that a sequence converges to a number a if its terms get arbitrarily close to a the further along in the sequence you get. The nautilus shell grows in the shape of a logarithmic spiral, which is closely related to the Fibonacci sequence. The Fibonacci sequences is an example of a recursively-defined sequence, because we can write it by the following recursive rule.īy the way, the Fibonacci sequence is important for many reasons, showing up in nature in the most unexpected ways. Here, the pattern is to start with two ones, and then to get each new term, we always add the previous two terms together. This is nothing more than the sequence of reciprocals of the natural numbers: a n = 1/ n. The formula for the general term is very simple: a n = n. You have probably seen and worked with many different kinds of sequences already even if you didn’t call them sequences. Moreover, if we know that a n = f( n) for some function f, then we say that f( n) is the formula for the general term. When n is unspecified, the expression a n is called the general term of the sequence. There are a number of different ways to write a sequence. Definition and NotationĪ sequence is a list of (infinitely many) numbers, called the terms of the sequence. Furthermore, we are often interested in determining whether a sequence converges (that is, approaches some fixed value) or not. We usually study infinite sequences, those that go on forever according to some rule or pattern. What are Sequences?īasically, a sequence is just a list of numbers. This review article is dedicated to sequences and their convergence properties. Just try always to make sure, whatever resource you're using, that you are clear on the definitions of that resource's terms and symbols.) In a set, there is no particular order to the elements, and repeated elements are usually discarded as pointless duplicates.One important topic that shows up on the AP Calculus BC exam (but not on the AB) is sequences. Unfortunately, notation doesn't yet seem to have been entirely standardized for this topic. (Your book may use some notation other than what I'm showing here. That is, they'll start at some finite counter, like i = 1.Īs mentioned above, a sequence A with terms a n may also be referred to as " ", but contrary to what you may have learned in other contexts, this "set" is actually an ordered list, not an unordered collection of elements. Infinite sequences customarily have finite lower indices. When a sequence has no fixed numerical upper index, but instead "goes to infinity" ("infinity" being denoted by that sideways-eight symbol, ∞), the sequence is said to be an "infinite" sequence. Don't assume that every sequence and series will start with an index of n = 1. Or, as in the second example above, the sequence may start with an index value greater than 1. This method of numbering the terms is used, for example, in Javascript arrays. The first listed term in such a case would be called the "zero-eth" term. Note: Sometimes sequences start with an index of n = 0, so the first term is actually a 0.
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