Given the infinite series ∑ n = 1 ∞ a n = a 1 + a 2 + a 3 + ⋯ ∑ n = 1 ∞ a n = a 1 + a 2 + a 3 + ⋯ and the corresponding sequence of partial sums { S k } where S k = ∑ n = 1 k a n = a 1 + a 2 + a 3 + ⋯ + a k , S k = ∑ n… To determine the convergence of a sequence given by an explicit formula a n = f ( n ) , we use the properties of limits for functions

If { a n } and { b n } are convergent sequences that converge to A and B , respectively, and c is any real number, then the sequence { c a n } converges to c · A , the sequences { a n ± b n } converge to A ± B , the… If a sequence is bounded and monotone, then it converges, but not all convergent sequences are monotone

sequence — an ordered list of numbers of the form a 1 , a 2 , a 3 ,… is a sequence. convergent sequence — a convergent sequence is a sequence { a n } for which there exists a real number L such that a n is arbitrarily close to L as long as n is sufficiently large. geometric sequence — a sequence { a n } in which the ratio a n + 1 / a n is the same for all positive integers n is called a geometric sequence. geometric series — a geometric series is a series that can be written in the form ∑ n = 1 ∞ a r n - 1 = a + a r 2 + a r 3 + ⋯ ∑ n = 1 ∞ a r n - 1 = a + a r 2 + a r 3 + ⋯.