Mathematics » Algebra » Sequences and Series
Sequences and series are fundamental concepts in algebra, serving as the building blocks for understanding various mathematical structures and their behaviors.
Sequences:
A sequence is essentially an ordered list of numbers governed by a specific rule or formula. Each number in the sequence is called a term. A sequence can be finite, having a fixed number of terms, or infinite, going on indefinitely. The general form of a sequence is denoted as \( \{a_n\} \), where \( n \) represents the position of the term in the sequence, and \( a_n \) is the value of the term at position \( n \).
One common type of sequence is the arithmetic sequence, where each term after the first is obtained by adding a constant difference, \( d \), to the preceding term. The \( n \)-th term of an arithmetic sequence can be expressed as:
\[ a_n = a_1 + (n-1)d \]
where \( a_1 \) is the first term.
Another important type is the geometric sequence, where each term after the first is obtained by multiplying the preceding term by a constant ratio, \( r \). The \( n \)-th term of a geometric sequence is given by:
\[ a_n = a_1 \cdot r^{n-1} \]
where \( a_1 \) is the first term.
Series:
When the terms of a sequence are summed together, the result is called a series. A series can also be finite or infinite, depending on whether the sequence has a fixed number of terms or continues indefinitely.
The sum of the first \( n \) terms of an arithmetic series can be found using the formula:
\[ S_n = \frac{n}{2} (a_1 + a_n) \]
or equivalently,
\[ S_n = \frac{n}{2} [2a_1 + (n-1)d] \]
For a geometric series, the sum of the first \( n \) terms is:
\[ S_n = a_1 \frac{1-r^n}{1-r} \quad \text{(for } r \neq 1\text{)} \]
If the geometric series is infinite and \( |r| < 1 \), the sum converges to:
\[ S = \frac{a_1}{1-r} \]
Understanding sequences and series is critical in various fields of mathematics, including calculus, where the concepts are extended to more complex functions and used in the study of convergence, limits, and continuous functions. They also play significant roles in computer science, physics, and finance, highlighting their interdisciplinary importance.