Many quantities that arise in applications cannot be computed exactly. We cannot write down an exact decimal expression for the number π or for values of the sine function such as sin 1. However, sometimes these quantities can be represented as infinite sums. For example, using Taylor series (Section 11.7), we can show that

Infinite sums of this type are called infinite series.

What precisely does Eq. (1) mean? It is impossible to add up infinitely many numbers, but what we can do is compute the partial sums SN, defined as the finite sum of the terms up to and including Nth term. Here are the first five partial sums of the infinite series for sin 1:

Compare these values with the value obtained from a calculator:

We see that S5 differs from sin 1 by less than 10−9. This suggests that the partial sums converge to sin 1, and in fact, we will prove that

(Example 2 in Section 11.7). So although we cannot add up infinitely many numbers, it makes sense to define the sum of an infinite series as a limit of partial sums.In general, an infinite series is an expression of the form

where {an} is any sequence. For example,

sin1 lim NNS

The Nth partial sum SN is the finite sum of the terms up to and including aN:

DEFINITION Convergence of an Infinite Series An infinite series

converges to the sum S if its partial sums converge to S:

In this case, we write

• If the limit does not exist, we say that the infinite series diverges.• If the limit is infinite, we say that the infinite series diverges to infinity.

Telescoping Series Investigate numerically:

Then compute the sum S using the identity:

:Numerically

The values of the partial sums suggest convergence to S = 1. To prove this, we observe that because of the identity, each partial sum collapses down to just two terms:

1

2

3

1 11 2

1 1 1 1 111 2 2 3 31 1 1 1 1 1 111 2 2 3 3 4 4

S

S

S

Telescoping Series Investigate numerically:

Then compute the sum S using the identity:

The values of the partial sums suggest convergence to S = 1. To prove this, we observe that because of the identity, each partial sum collapses down to just two terms:

111NS

N

1lim lim 1 11NN N

S SN

In most cases (apart from telescoping series and the geometric series introduced below), there is no simple formula like Eq. (2) for the partial sum SN. Therefore, we shall develop techniques that do not rely on formulas for SN.

It is important to keep in mind the difference between a sequence {an} and an infinite series

Sequences versus Series Discuss the difference between {an} and

1

1, where .1n n

n

a an n

The sequence is the list of numbers

This sequence converges to zero:

The infinite series is the sum of the numbers an, defined formally as the limit of the partial sums. This sum is not zero. In fact, the sum is equal to 1 by Example 1:

1 1 1, , , ....

1 2 2 3 3 4

1lim lim 0

1nn na

n n

1 1

1 1 1 1 11 1 2 2 3 3 4n

n n

an n

Make sure you understand the difference between sequences and series.

• With a sequence, we consider the limit of the individual terms an.• With a series, we are interested in the sum of the terms a1 + a2 + a3 +… which is defined as the limit of the partial sums.

THEOREM 1 Linearity of Infinite Series

and converge and can also

converge ( is any constant), andn n n n na b a b ca

c

, is any constant

n n n n

n n n n

n n

a b a b

a b a b

ca c a c

A main goal in this chapter is to develop techniques for determining whether a series converges or diverges. It is easy to give examples of series that diverge:

• diverges to infinity (the partial sums increase without bound):

S1 = 1, S2 = 1 + 1 = 2, S3 = 1 + 1 + 1 = 3, S4 = 1 + 1 + 1 + 1 = 4,…

• diverges (the partial sums jump between 1 and 0):

S1 = 1, S2 = 1 − 1 = 0, S3 = 1 − 1 + 1 = 1, S4 = 1 − 1 + 1 − 1 = 0,…

1

1n

S

1

1

1 n

n

S

Next, we study the geometric series, which converge or diverge depending on the common ratio r.

A geometric series with common ratio r 0 is a series defined by a geometric sequence crn, where c 0. If the series begins at n = 0, then

For and c = 1, we can visualize the geometric series starting at n = 1 (Figure 1):

1/ 2r

Adding up the terms corresponds to moving stepwise from 0 to 1, where each step is a move to the right by half of the remaining distance. Thus S = 1.

2 3 4 5

0

n

n

S cr c cr cr cr cr cr

1

1 1 1 1 1 12 2 4 8 16n

n

S

2 3

2 3 1

1

11 1

NN

N NN

NN N

NN

S c cr cr cr cr

rS cr cr cr cr cr

S rS c cr

S r c r

There is a simple device for computing the partial sums of a geometric series:

If r 1, we may divide by (1 − r) to obtain

This formula enables us to sum the geometric series.

12 3

1

1

NN

N

c rS c cr cr cr cr

r

THEOREM 2 Sum of a Geometric Series Let c 0. If |r| < 1, then

If |r| ≥ 1, then the geometric series diverges.

2 3

0 1n

n

ccr c cr cr crr

12 3

1

1

NN

N

c rS c cr cr cr cr

r

1 2 3

1

Mn M M M M

n M

crcr cr cr cr crr

0

Evaluate 5 .n

n

0 0

1 1 11 1 & 55 5 1

5/ 44 5

nn

n n

cc rr

Evaluate

3

3

373 474 1 7 6/ 4

271

n M

n

crr

2 3

0 1n

n

ccr c cr cr crr

1 2 3

1

Mn M M M M

n M

crcr cr cr cr crr

0

2 3Evaluate .5

n

nn

S

Write S as a sum of two geometric series. This is valid by Theorem 1 because both geometric series converge:

0 0 0

1 3 1 32 25 5 5 5

2 1 5 54 /

55 2 / 5 2 2

n n n n

n n n

S

CONCEPTUAL INSIGHT Sometimes, the following incorrect argument is given for summing a geometric series:

Thus, 2S = 1 + S, or S = 1. The answer is correct, so why is the argument wrong? It is wrong because we do not know in advance that the series converges. Observe what happens when this argument is applied to a divergent series:

This would yield 2S = S − 1, or S = −1, which is absurd because S diverges. We avoid such erroneous conclusions by carefully defining the sum of an infinite series as the limit of partial sums.

1 1 1 2 4 81 1 12 1 12 4 8

S

S S

1 2 4 8 162 2 4 8 16 1S

S S

The infinite series

diverges because the Nth partial sum SN = N diverges to infinity. It is less clear whether the following series converges or diverges:

We now introduce a useful test that allows us to conclude that this series diverges.

THEOREM 3 Divergence Test If the nth term an does not converge to zero, then the series

1

diverges.nn

a

The Divergence Test (also called the nth-Term Test) is often stated as follows:

1

converges lim 0n nnn

a a

Prove the divergence of 1

.4 1n

nSn

1lim diverges.

4nna S

Determine the convergence or divergence of

11, & 1 diverges diverges1

nn Sn

The Divergence Test tells only part of the story. If an does not tend to zero, then certainly diverges. But what if an does tend to zero? In this case, the series may converge or it may diverge. In other words, is a necessary condition of convergence, but it is not sufficient. As we show in the next example, it is possible for a series to diverge even though its terms tend to zero.

Sequence Tends to Zero, yet the Series Diverges Prove the divergence of

1/ 0, butN because each term in the sum SN is greater than or equal to 1/ , we haveN

This shows that increases without bound (Figure 2). Therefore SN also increases without bound. This proves that the series diverges.

. But NS N N

Geometric series were used as early as the third century BCE by Archimedes in a brilliant argument for determining the area S of a “parabolic segment” (shaded region in Figure 3). Given two points A and C on a parabola, there is a point B between A and C where the tangent line is parallel to AC(apparently, Archimedes knew the Mean Value Theorem more than 2000 years before the invention of calculus). Let T be the area of triangle ΔABC. Archimedes proved that if D is

chosen in a similar fashion relative to AB and E is chosen relative to ,BC then

This construction of triangles can be continued. The next step would be to construct the four triangles on the segments of total area . Then construct eight triangles of total area , etc. In this way, we obtain infinitely many triangles that completely fill up the parabolic segment. By the formula for the sum of a geometric series,

For this and many other achievements, Archimedes is ranked together with Newton and Gauss as one of the greatest scientists of all time. The modern study of infinite series began in the seventeenth century with Newton, Leibniz, and their contemporaries. The divergence of

(called the harmonic series) was known to the medieval scholar Nicole d’Oresme (1323–1382), but his proof was lost for centuries, and the result was rediscovered on more than one occasion. It was also known that the sum of the reciprocal squares converges, and in the 1640s,

the Italian Pietro Mengoli put forward the challenge of finding its sum. Despite the efforts of the best mathematicians of the day, including Leibniz and the Bernoulli brothers Jakob and Johann, the problem resisted solution for nearly a century. In 1735, the great master Leonhard Euler (at the time, 28 years old) astonished his contemporaries by proving that

This formula, surprising in itself, plays a role in a variety of mathematical fields. A theorem from number theory states that two whole numbers, chosen randomly, have no common factor with probability 6/π2 ≈ 0.6 (the reciprocal of Euler’s result). On the other hand, Euler’s result and its generalizations appear in the field of statistical mechanics.

Archimedes (287 BCE–212 BCE), who discovered the law of the lever, said “Give me a place to stand on, and I can move the earth” (quoted by Pappus of Alexandria c. AD 340).

Archimedes showed that the area S of the parabolic segment iswhere T is the area of ΔABC.