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Find a formula for the general term $ a_n $ of the sequence, assuming that the pattern of the first few terms continues.$ \left\{\begin{array} \frac {1}{2}, - \frac {4}{3}, \frac {9}{4}, - \frac {16}{5}, \frac {25}{6}, . . . . .\end{array}\right\} $

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$a_{n}=\frac{(-1)^{n-1} n^{2}}{n+1}$

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 1

Sequences

Series

University of Michigan - Ann Arbor

University of Nottingham

Boston College

Lectures

01:59

In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a finite sequence of real numbers is called a finite series. The sum of an infinite sequence of real numbers may or may not have a well-defined sum, and may or may not be equal to the limit of the sequence, if it exists. The study of the sums of infinite sequences is a major area in mathematics known as analysis.

02:28

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers (for infinite sequences) or the set of the first "n" natural numbers (for a finite sequence). A sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations and analysis. Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in the study of prime numbers.

01:46

Find a formula for the gen…

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find a formula for the gen…

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01:40

The first several terms of…

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00:28

Find a formula for the nth…

that first term supposed to be one half can't one thing that we notices that there's in the denominator all devising the denominator, always just increasing by one. They're always going to be increasing by one. And we need to add one here so that we start on, too. We should start on too. Okay, So if we have in plus one in the denominator, that means that we'LL start on too. But we'LL be one more. Every time I write this in, he's going to make us be one more in the previous term, every time in the denominator, and we're also going to be alternating signs. First term is positive. Next term is negative. Next term is positive. It's where all training signs so well, maybe something like this. Well, then you can check to see if you plug in and equals one. Then you'd have minus one to the one, which would give you a negative number. So then you need to modify that. If you put in minus one there, then you should start on a positive number, which is what you want, Kane. And the only other thing that we need is that we have powers and the numerator. So the first term we have one squared. Next term we have ah, four, which is two squared. The next term we have a nine, which is three squared. Next term we have a sixteen, which is four squared. So we should have in in squared here. Okay, so a n should be minus one to the end, minus one times and squared, divided by N plus one.

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