What is the difference between a sequence and a series?
Ah! You’ve found the point of most initial difficulty concerning infinite series. At the point in the course where we have not yet talked about infinite series, but only aboput sequences, I’ll give a preview. An infinite series is written in such a way as to look like a sum of infinitely many terms. Since we can’t really add infinitely many terms by ordinary addition, we “work around” that by creating a sequence. This sequence is called the sequence of partial sums. The sequence of partial sums has 1st term that is the same as the 1st term of the sum (series), 2nd term that is the sum of the first 2 terms of the series, 3rd term that is the sum of the first3 terms of the series, and so on. See p. 714 of Stewart. SERIES CONVERGENCE 20.Theorem 6 says that if an infinite series is convergent, then its nth term must have a limit of 0; this means that if the limit of the nth term of a series is not 0, then the series is divergent. So how exactly can we know that a series is convergent when
