Notes on Infinite Sequences and Series

We represent a generic sequence as a 1 , a 2 , a 3 ,. .. , and its n-th as a n. In order to define a sequence we must give enough information to find its n-th term. Two ways of doing this are: 1. With a formula. E.g.: a n = 1 n a n = 1 10 n a n = √ 3n − 7 2. With a recursive definition. 1.2. Limit of a Sequence. We say that a sequence a n converges to a limit L if the difference |a n − L| can be made as small as we wish by taking n large enough. We write a n → L, or more formally: lim n→∞ a n = L. lim n→∞ 1 n = 0 .