SECTION 8.2 SERIES

P2P28.2 SERIES  If we try to add the terms of an infinite sequence we get an expression of the form a 1 + a 2 + a 3 + ··· + a n + ···

P3P38.2 INFINITE SERIES  This is called an infinite series (or just a series). It is denoted, for short, by the symbol.  However, does it make sense to talk about the sum of infinitely many terms?

P4P48.2 INFINITE SERIES  It would be impossible to find a finite sum for the series ··· + n + ··· If we start adding the terms, we get the cumulative sums 1, 3, 6, 10, 15, 21,... After the nth term, we get n(n + 1)/2, which becomes very large as n increases.

P5P58.2 INFINITE SERIES  However, if we start to add the terms of the series we get:

P6P68.2 INFINITE SERIES  The table shows that, as we add more and more terms, these partial sums become closer and closer to 1. In fact, by adding sufficiently many terms of the series, we can make the partial sums as close as we like to 1.

P7P78.2 INFINITE SERIES  So, it seems reasonable to say that the sum of this infinite series is 1 and to write:  We use a similar idea to determine whether or not a general series (Series 1) has a sum.

P8P88.2 INFINITE SERIES  We consider the partial sums s 1 = a 1 s 2 = a 1 + a 2 s 3 = a 1 + a 2 + a 3 s 3 = a 1 + a 2 + a 3 + a 4 In general,

P9P98.2 INFINITE SERIES  These partial sums form a new sequence {s n }, which may or may not have a limit.  If exists (as a finite number), then, as in the preceding example, we call it the sum of the infinite series Σa n.

P108.2 Definition 2  Given a series, let s n denote its nth partial sum:  If the sequence {s n } is convergent and exists as a real number, then the series Σa n is called convergent and we write: The number s is called the sum of the series. Otherwise, if the sequence {s n } us divergent, then the series is called divergent.

P118.2  Thus, the sum of a series is the limit of the sequence of partial sums. So, when we write, we mean that, by adding sufficiently many terms of the series, we can get as close as we like to the number s.  Notice that: SUM OF INFINITE SERIES

P128.2 SUM OF INFINITE SERIES VS. IMPROPER INTEGRALS  Compare with the improper integral To find this integral, we integrate from 1 to t and then let t → ∞. For a series, we sum from 1 to n and then let n → ∞.

P138.2 Example 1  An important example of an infinite series is the geometric series  Each term is obtained from the preceding one by multiplying it by the common ratio r. We have already considered the special case where a = ½ and r = ½ earlier in the section.

P148.2 GEOMETRIC SERIES  If r = 1, then s n = a + a + … + a = na → ±∞ Since doesn ’ t exist, the geometric series diverges in this case.  If r ≠ 1, we have s n = a + ar + ar 2 + … + ar n – 1 and rs n = ar + ar 2 + … +ar n – 1 + ar n

P158.2 GEOMETRIC SERIES  Subtracting these equations, we get: s n – rs n = a – ar n  If – 1 < r < 1, we know from Result 8 in Section 8.1 that r n → 0 as n → ∞.

P168.2  So, Thus, when |r | < 1, the series is convergent and its sum is a/(1 – r).  If r ≤ – 1 or r > 1, the sequence {r n } is divergent by Result 8 in Section 8.1  So, by Equation 3, does not exist. Hence, the series diverges in those cases. GEOMETRIC SERIES

P178.2 GEOMETRIC SERIES  Figure 1 provides a geometric demonstration of the result in Example 1.  If s is the sum of the series, then, by similar triangles,  So,

P188.2 The geometric series is convergent if |r | < 1 and the sum of the series is: If |r | ≥ 1, the series is divergent. GEOMETRIC SERIES  We summarize the results of Example 1 as follows.