INFINITE SEQUENCES AND SERIES

12 INFINITE SEQUENCES AND SERIES

INFINITE SEQUENCES AND SERIES 12.2Series • 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. p. 724

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 Σan is 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. Fig. 12.2.1, p. 725

GEOMETRIC SERIES • If s is the sum of the series, then, by similar triangles, • So, Fig. 12.2.1, p. 725

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. p. 726 Fig. 12.2.2, p. 726

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. Fig. 12.2.3, p. 727

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