Introduction to Sequences and Series
In mathematics, a sequence or series (i.e. a finite or infinite collection of numbers) is an ordered list of elements. A sequence can be thought of as a list in which each subsequent term is determined by the sequence's previous term (the rules of the sequence). A series is a list of numbers with a defined order of terms (a common order is ascending order). A sequence's terms may be numbers, but may also be other mathematical objects such as functions or sets. In mathematics, the terms order of magnitude and magnitude are used to describe the size of a number. In particular, the term order of magnitude is used to describe the size of a number in relation to the numbers it is compared to. For instance, the order of magnitude of 2 is 1, since it can be written as 2 = 2×1. The term magnitude, on the other hand, is a term used to describe a number in relation to its unit, for instance, the magnitude of 2 is 2. A number that is one less than its order of magnitude is, by definition, equal to its order of magnitude. For example, the order of magnitude of 1,000 is 100, since it can be expressed as 1,000 = 100×100. The magnitude of 1,000 is 10,000, since it can be expressed as 1,000 = 10,000. Therefore, the magnitude of 1,000 is equal to 10. Some authors call the magnitude of a number the "approximate value" of the number. In mathematics, an infinite sequence or series of numbers is an infinite list (or one that can potentially expand to an infinite list) of numbers. In mathematics, sequences and series are often treated as the same concept, but a series is generally considered to be a finite sequence whose terms are considered to be numbers. In contrast, a sequence has no end and always has a next term. For instance, the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ... is infinite, since it is possible for the sequence to produce an infinite series of numbers. In contrast, the series 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... is finite, since it can never produce an infinite series of numbers. This is because the series only has up to 10 terms. In general, a series is the sum of a sequence of terms, and a sequence is the difference of the terms of a series: In this case, the term "series" is shortened to "s". The series sum can be written as s = a1 + a2 + a3 ... + an = (a1 + a2 + a3 + ... + an) + an For example, the series 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ... can be written as 1 + 1 + 2 + 3 + 5 + ... + n, where each number is the sum of the previous n terms. Therefore, the series sum is the sum of the sequence of terms. The sequence difference can be written as s = a1 – a2 – a3 ... – an – an For example, the sequence 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... can be written as 1 – 2 – 3 – 4 – 5 – 6 – 7 – 8 – 9 – 10, where each number is the difference between the previous n terms. Therefore, the sequence difference is the difference between the sequence of terms. By definition, a sequence is a finite or infinite list of terms. A series can be formed by taking the difference of a sequence; this is a special case of the general mathematical definition of a sequence as an infinite list of terms. A sequence of numbers can be written in the form a1, a2, a3, ..., an, with the subscripts 1, 2, 3, ..., n indicating the sequence of numbers, and "a1" the first number of the sequence. In the case of a finite sequence, the terms are finite numbers. For example, the sequence of integers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ... is a sequence of the natural numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ... (with a gap of one between the numbers 6 and 7). A sequence of real numbers can be written a1, a2, a3, ..., an, where "a1", "a2", "a3", ..., "an" are the partial sums of the sequence. A sequence can be defined
