INFINITE SEQUENCES AND SERIES

11 INFINITE SEQUENCES AND SERIES

INFINITE SEQUENCES AND SERIES 11.2Series In this section, we will learn about: Various types of series.

SERIES Series 1 If we try to add the terms of an infinite sequence we get an expression of the form a1 + a2 + a3 + ··· + an + ∙·∙

INFINITE SERIES This is called an infinite series (or just a series). • It is denoted, for short, by the symbol

INFINITE SERIES However, does it make sense to talk about the sum of infinitely many terms?

INFINITE SERIES It would be impossible to find a finite sum for the series 1 + 2 + 3 + 4 + 5 + ∙∙∙ + 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.

INFINITE SERIES However, if we start to add the terms of the series we get:

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.

INFINITE SERIES So, it seems reasonable to say that the sum of this infinite series is 1 and to write:

INFINITE SERIES We use a similar idea to determine whether or not a general series (Series 1) has a sum.

INFINITE SERIES We consider the partial sums s1 = a1 s2 = a1 + a2 s3 = a1 + a2 + a3 s3 = a1 + a2 + a3 + a4 • In general,

INFINITE SERIES These partial sums form a new sequence {sn}, which may or may not have a limit.

SUM OF INFINITE SERIES If exists (as a finite number), then, as in the preceding example, we call it the sum of the infinite series Σan.

SUM OF INFINITE SERIES Definition 2 Given a series let sn denote its nth partial sum:

SUM OF INFINITE SERIES Definition 2 If the sequence {sn} is convergent and exists as a real number, then the series Σanis called convergentand we write: • The number s is called the sumof the series. • Otherwise, the series is called divergent.

SUM OF INFINITE SERIES 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.

SUM OF INFINITE SERIES Notice that:

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

GEOMETRIC SERIES Example 1 An important example of an infinite series is the geometric series

GEOMETRIC SERIES Example 1 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.

GEOMETRIC SERIES Example 1 If r = 1, then sn = a + a + ∙∙∙ + a = na → ±∞ • Since doesn’t exist, the geometric series diverges in this case.

GEOMETRIC SERIES Example 1 If r≠ 1, we have sn = a + ar + ar2 + ∙∙∙ + ar n–1 and rsn = ar + ar2 + ∙∙∙ +ar n–1 + ar n

GEOMETRIC SERIES E. g. 1—Equation 3 Subtracting these equations, we get: sn – rsn = a – ar n

GEOMETRIC SERIES Example 1 If –1 < r < 1, we know from Result 9 in Section 11.1 that r n → 0 as n → ∞. So, • Thus, when |r | < 1, the series is convergent and its sum is a/(1 – r).

GEOMETRIC SERIES Example 1 If r≤ –1 or r > 1, the sequence {r n} is divergent by Result 9 in Section 11.1 So, by Equation 3, does not exist. • Hence, the series diverges in those cases.

GEOMETRIC SERIES The figure provides a geometric demonstration of the result in Example 1.

GEOMETRIC SERIES If s is the sum of the series, then, by similar triangles, So,

GEOMETRIC SERIES We summarize the results of Example 1 as follows.

GEOMETRIC SERIES Result 4 The geometric series is convergent if |r | < 1.

GEOMETRIC SERIES Result 4 The sum of the series is: If |r | ≥ 1, the series is divergent.

GEOMETRIC SERIES Example 2 Find the sum of the geometric series • The first term is a = 5 and the common ratio is r = –2/3

GEOMETRIC SERIES Example 2 • Since |r | = 2/3 < 1, the series is convergent by Result 4 and its sum is:

GEOMETRIC SERIES What do we really mean when we say that the sum of the series in Example 2 is 3? • Of course, we can’t literally add an infinite number of terms, one by one.

GEOMETRIC SERIES However, according to Definition 2, the total sum is the limit of the sequence of partial sums. • So, by taking the sum of sufficiently many terms, we can get as close as we like to the number 3.

GEOMETRIC SERIES The table shows the first ten partial sums sn. The graph shows how the sequence of partial sums approaches 3.

GEOMETRIC SERIES Example 3 Is the series convergent or divergent?

GEOMETRIC SERIES Example 3 Let’s rewrite the nth term of the series in the form arn-1: • We recognize this series as a geometric series with a = 4 and r = 4/3. • Since r > 1, the series diverges by Result 4.

GEOMETRIC SERIES Example 4 Write the number as a ratio of integers. • 2.3171717… • After the first term, we have a geometric series with a = 17/103 and r = 1/102.

GEOMETRIC SERIES Example 4 • Therefore,

GEOMETRIC SERIES Example 5 Find the sum of the series where |x| < 1. • Notice that this series starts with n = 0. • So, the first term is x0 = 1. • With series, we adopt the convention that x0 = 1 even when x = 0.

GEOMETRIC SERIES Example 5 Thus, • This is a geometric series with a = 1 and r = x.

GEOMETRIC SERIES E. g. 5—Equation 5 Since |r | = |x| < 1, it converges, and Result 4 gives:

SERIES Example 6 Show that the series is convergent, and find its sum.

SERIES Example 6 This is not a geometric series. • So, we go back to the definition of a convergent series and compute the partial sums:

SERIES Example 6 We can simplify this expression if we use the partial fraction decomposition. • See Section 7.4

SERIES Example 6 Thus, we have:

SERIES Example 6 Thus, • Hence, the given series is convergentand

SERIES The figure illustrates Example 6 by showing the graphs of the sequence of terms an=1/[n(n + 1)] and the sequence {sn} of partial sums. • Notice that an → 0 and sn → 1.

HARMONIC SERIES Example 7 Show that the harmonic series is divergent.

HARMONIC SERIES Example 7 For this particular series it’s convenient to consider the partial sums s2, s4, s8, s16, s32,…and show that they become large.

INFINITE SEQUENCES AND SERIES