If you get stuck at this step, imagine how you would handle the problem if it involved x instead of n. (b) We need to compute the limit as n approaches infinity.
(a) Each term of a sequence can be found by plugging in n-values, similar to how you might find the value of a function f( x) by plugging in x-values. If it converges, then what is the limit? Solution (b) Determine whether the sequence converges. (a) Write out the first four terms of the sequence. 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. One important topic that shows up on the AP Calculus BC exam (but not on the AB) is sequences.
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