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Series and sequences math and science initiative murphy
Series and sequences math and science initiative murphy










series and sequences math and science initiative murphy

We can "remove" terms from the above series as follows: It is also possible to "remove" terms from the series in a manner that can be (very loosely) related to factoring. This will always be true: if the index is increased by some value x, n will decrease by the same value x on the other hand, if the index is decreased by some value x, n will increase by the same value x.

series and sequences math and science initiative murphy

In the above examples, we increased the initial value of the index by 1 each time, which resulted in all the n's in the series decreasing by 1. Similarly, we could shift the starting index to n = 2, and the resulting equivalent series would be as follows: If we instead wanted to begin the series at n = 1, we can shift the index as follows: In the above series, the starting index is n = 0. In some cases, given some series and starting index, it is useful to shift the starting index so as to begin the series at a different value. Instead, the product of two series can be written as,Īlthough the index used is often n = 0 or n = 1, it is important to note that the index can start at any n. It is worth noting that series cannot be multiplied in the same way as they are added. Given two convergent sequences, a n and b n, their series can be added or subtracted as follows: Given some constant c and sequence a n, we can factor the constant out of the series as follows: It is always possible to factor a multiplicative constant out of a series. There are a number of properties of series that can be used to manipulate series and thereby simplify or alter them in useful ways. When the above limit is equal to some real number S, the limit of the partial sums of the sequence, and therefore the series, converges. Then, the limit of the partial sums can be expressed in series notation as: When it is convergent, the series is said to be summable (specifically the sequence is summable), and a value can be assigned by computing the limit of the partial sums of the sequence, where the partial sum of a sequence may be defined as follows: When a series is divergent, the sum of the series cannot be computed.

series and sequences math and science initiative murphy

This reads as "the sum from n equals one to infinity of a sub n."Īs mentioned above, series can be either convergent or divergent. , where the subscript denotes which term in the series is being represented.Series are commonly represented in two ways: as a sum of variables followed by an ellipsis (.) or with the use of the summation sign, as shown below:

series and sequences math and science initiative murphy

Since finite series are not usually considered in the context of calculus, any use of the word "series" on this site will mean infinite series unless otherwise specified. The "infinite" in infinite series is meant to emphasize that the series contains an infinite number of terms. "Series" and "infinite series" are often used interchangeably. Terms can be numbers, functions, or essentially anything that can be added. The values of a sequence, the sum of which form a series, are referred to as terms or elements. In order to work with series, it is important to have an understanding of the notation used. Series are used throughout many different fields of study including mathematics (particularly calculus), physics, computer science, statistics, finance, and more. When a series diverges, it means that the sum either does not exist or is ±∞. When a series converges, it is a single value, since it is the sum of an infinite sequence. An infinite series is the sum of an infinite sequence. In mathematics, the term series is typically used to describe an infinite series.












Series and sequences math and science initiative murphy