Sequences and Series

A geometric sequence is a sequence in which the ratio of any term to the previous term is constant. The explicit formula for a geometric sequence is of the form a_n = a₁^r-1, where r is the common ratio. A geometric sequence can be defined recursively by the formulas a₁ = c, a_n+1 = ra_n, where c is a constant and r is the common ratio.

The sum of a finite geometric sequence (the value of a geometric series) can be found according to a simple formula. For a geometric sequence a_n = a₁r^n-1, the sum of the first n terms is S_n = a₁(.

In calculus, the study of infinite geometric series is very involved. Deciding whether an infinite geometric series is convergent or divergent, and finding the limits of infinite geometric series are only two of many topics covered in the study of infinite geometric series. In this text, we'll only use one formula for the limit of an infinite geometric series. In the case in which -1 < r < 1, the limit of the infinite geometric series a₁r^n-1 = . This limit of the series is the same as the sum of the infinite geometric sequence a_n = a₁r^n-1.