Series

The sum of the terms of a sequence is called a series. Example Sequence:& 2 , 4 , 6 , 8 Example Series:& 2 + 4 + 6 + 8 Depending on the number of terms, a series can be finite or infinite. A finite series has a finite number of terms. This means that there is a first and last term. On the other hand, an infinite series has an infinite number of terms. Therefore, there is a first term but not a last term because the terms continue to infinity. Example Finite Series 2 + 4 + 6 + 8 [1em] Example Infinite Series 2 + 4 + 6 + 8 + 10 + ⋯ If the terms of a sequence are given by a rule, it is usually more compact to write the corresponding series using sigma notation. The sequence above is described by the explicit rule a_n=2n. Therefore, the series can be written using sigma notation. 2 + 4 + 6 + 8=∑_(n=1)^4 2n Here, by substituting n with integers from 1 (under Σ) through 4 (above Σ), the individual terms of the series are obtained. The symbol Σ itself indicates that between each term there is an addition sign. This alternative notation is also useful for infinite series. In this case, the infinity symbol is written above Σ. 2 + 4 + 6 + 8 + 10 + ⋯=∑_(n=1)^(∞) 2n

If the sum of an infinite series approaches a number as n tends to infinity, then the series is said to converge to that number. Otherwise, the series diverges. Example Convergent Series ∑_(n=1)^(∞) 1/2^n =1/2 + 1/4 + 1/8 + 1/16+ ⋯ = 1 [1.5em] Example Divergent Series ∑_(n=1)^(∞) 3^n=3+9+27+ ⋯ → ∞