**Chapter 13**

June 03 2017

**Chapter 13 Introduction to Infinite Series**
**13.1 What is an Infinite Series?**
- an infinite series
- or simply a series
- is an expression of the form
- a1 + a2 + a3 + ...
- where the terms continue indefinitely
- we will often use the sigma notation of section 6.3 to write the series in the form
- E1inf an
- it was one of the greatest achievements of the nineteenth century mathematics to discover
- that a perfectly reasonable and satisfactory meaning can be given to our infinite series
- if we exercise suitable caution
- this meaning allows us to work with such expressions just as easily as if they involved only a finite number of terms
- in many cases we will actually be able to find the number that is the exact sum of the infinite series
- and these sums often turn out to be very surprising indeed
- first we briefly consider a few of the many natural ways in which infinite series arise in mathematics
- an infinite decimal is defined in a purely numerical way as an infinite series
- 1/3 = 0.333 ...
- the elementary long division 1/1-x = 1 + x + x2 + ...
- the binomial theorem (1+x)n expands as an infinite series if n is a fraction
- infinite series arise in a very insistent way in the study of differential equations
- dy/dx = y
- y = a0 (1 + x + x2/2! + ...)
- ex = 1 + x + x2/2! + ...
- for all values of x we can calculate ex as accurately as we please from the series by taking enough terms
- this is how numerical tables for ex are constructed
- in attempting to solve other differential equations in this way
- we are led to other series
- for example, the differential equation
- x d2y/dx2 + dy/dx + xy = 0
- many applications to mathematics physics
- one of its solutions is an infinite series
- as in almost any part of calculus
- there are certain things we want to be able to do freely with the tools we are studying
- the role of the theory is mostly to justify and legitimate the various procedures that are necessary for carrying out our purpose
**13.2 The Convergence and Divergence of Series**
- the idea of infinite addition arises in its most transparent form in the famous formula
- 1 + 1/2 + 1/4 + ... = 2
- the reason is it produces the sequence of partial sums
- 1, 1 1/2, 1 3/4 ...
- visibly approaching 2 as its limit
- this is an example of a geometric series
- to form such a series
- we start with a number a != 0
- and a second number r between -1 and 1
- we construct the geometric progression
- a, ar, ar2 ...
- r is called the ratio
- our purpose is to calculate the sum of the geometric series formed from this progression
- the finite sum of the terms from a to arn is called the nth partial sum, sn
- if the series has an exact sum s
- it certainly seems reasonable that we should approach closer to s as we add more terms
- so s = lim n>inf sn
- what makes an explicit calculation possible
- is the fact that there exists a simple closed formula for the nth partial sum sn
- we multiply sn by r and write the two sums together
- these sums share many terms in common
- so sn - rsn = a - ar n+1
- sn = a(1 - r n+1) / 1-r
- since r != 1
- r n+1 > 0 as n > inf
- because |r| < 1
- so sn > a / 1-r as n > inf
- so assuming that there is an exact sum s for the geometric series
- s = lim n>inf sn = a / 1-r
- amazingly
- 0.333 ... = 3/10 + 3/10 2 + ... = 3/10 / 1 - 1/10 = 1/3
- we now look back on the ideas discussed above
- and introduce a small but significant change in our point of view
- further investigation will not uncover any way to calculate the exact sum of the geometric series other than as the limit s = lim sn
- so instead of simply assuming that the series has a sum
- and that our job is only to calculate it
- it is logically better to reverse our approach and define the sum of the series a + ar + ...
- to be the number calculated this way
- which we know exists by the above discussion
- this altered point of view
- is extremely important for clarifying our understanding of infinite series in general
- because, as we shall see
- some series have sums and others do not
- these ideas provide the pattern for our broader study of infinite series
- and the sum of any other series
- a1 + a2 + ...
- is defined and calculated in the same way
- so we begin by forming the sequence of partial sums
- sn = a1 + ... + an
- if this sequence of longer and longer partial sums approaches a finite limit s
- then we say the series converges to s
- we write a + ... + an = s
- and call s the sum of the series
- the conclusion we have reached about the geometric series can now be expressed as follows
- if |r| < 1 then the geometric series
- a + ar + ar2 + ...
- converges and its sum is a / 1-r
- when an infinite series fails to have a sum
- we say that it diverges
- this means the partial sums sn
- fail to approach a finite limit as n > inf
- this can happen in several different ways
- for example 1 + 1 + ... diverges to inf
- the harmonic series 1 + 1/2 + 1/3 ... can be arranged to be 1/2 + 1/2 + ... so it also diverges to inf
- 1 - 1 + 1 ... oscillates
- a simple test for divergence that is often useful is the nth term test
- if the series E 1 inf an = a1 + a2 + ...
- converges
- then an > 0 as n > inf
- if an does not approach zero as n > inf
- then the series must diverge
- since both sn and sn - 1 are close to s when n is large
- they are close to each other
- so their difference, an, must be close to zero
- repeating decimals
- the procedure for converting any rational number a/b
- into its decimal expansion is well known
- divide and find remainders
- when remainders repeat
- the digits of the decimals repeat
- for example
- 22/7 is a good approximation for pi
- this illustrates and almost proves
- the decimal expansion of any rational number is repeating
- the proof of this general statement consists of little more than noticing the phenomenon displayed in the example
- when b is divided into a
- the remainder at each stage is one of the numbers 0, 1, 2 ... b-1
- since there are only a finite number of possible values for these remainders
- some remainder necessarily appear a second time
- and the division process repeats from that point on to give a repeating decimal sequence
- if the remainder 0 appears
- it can always be thought of as repeating
- the converse of this statement is also true
- any repeating decimal is the expansion of a rational number
- by constructing the repeating part as an infinite converging series
- repeating with powers of ten
- we can summarize our results by saying that the rational numbers are exactly those real numbers whose decimal expansions are repeating
- equivalently
- the irrational numbers are exactly those real numbers whose decimal expansions are non-repeating
- one of the attractions of the subject is that it offers so many results that stir the imagination and stimulate curiosity
- it seems reasonably clear that a series of positive terms will converge if its terms decrease rapidly enough
**13.3 Various Series Related to the Geometric Series**
- as we shall see in this section
- the geometric series
- 1 + x + x2 + ...
- E 0 inf x n
- is the most useful of all infinite series
- a series of functions cannot be said to converge or diverge as it stands
- since it is not an infinite series of numbers
- it becomes a series of numbers when we give x a numerical value
- in general we expect such a series will converge for some values of x and diverge for other values
- we can use the nth term test to conclude the series
- diverges if x >= 1 or x <= -1
- since xn does not approach 0
- the fact the series converges for all other values of x
- is easily seen from the formula for the nth partial sum
- sn = 1 + x + x2 ... = 1 - x n+1 / 1 - x
- because |x| < 1
- x n + 1 > 0 so sn > 1 / 1 - x
- the interval -1 < x < 1
- is called the interval of convergence
- this geometric series is a rare example of an infinite series
- whose sum we are able to find
- by first finding a simple formula for its nth partial sum sn
- part of the importance of this series
- lies in the fact that it can be used as a starting point
- for determining the sums of many other interesting series
- one way to do this is to replace x by various functions
- substituting in -1 < x < 1
- to find the interval of convergence for the new series
- replacing x with a power of x
- are known as power series
- and all have the special form
- E 0 inf an xn = a0 + a1 x + a2 x2 + ...
- the numbers a0, a1 are the coefficients of the power series
- the basic form of all a = 1 is the simplest power series and the prototype for all such series
- differentiating and integrating both sides sometimes can lead to suggestions for remarkable new formulas
- the logarithm series
- ln (1 + x) = x - x2 / 2 + x3 / 3 ...
- the series allows us to calculate the logarithm of any positive number
- the inverse tangent series
- tan -1 x = x - x3 / 3 + x5 / 5 ...
- Leibniz's formula
- pi / 4 = 1 - 1/3 + 1/5 ...
- follows at once at x = 1
- the computation of pi
- in principle
- Leibniz's formula can be used for computing the numerical value of pi
- but practically
- the series converges too slowly
- alternative more efficient methods can be used
**13.4 Power Series Considered Informally**
- polynomials are the simplest functions of all
- power series can be thought of polynomials of infinite degree
- an expression of a function f(x) in a power series
- f(x) = a0 + a1 x + ... + an xn + ...
- is a way of expressing f(x) by means of functions of a particularly simple kind
- our investigations in the preceding sections have led us to the following power series expressions, among others:
- 1/1+x = 1 - x + x2 - x3 + ...
- ln (1 + x) = x - x2/2 + x3/3 - ...
- tan -1 x = x - x3/3 + x5/5 - ...
- ex = 1 + x + x2/2! + x3/3! + ...
- sin x = x - x3/3! + x5/5! - ...
- cos x = 1 - x2/2! + x4/4! - ...
- these power series expansions are among the most important formulas in all of mathematics
- these formulas show that many different types of functions can be expanded in power series
- these formulas are all special cases of a very general and powerful formula that enable us to find the unique power series expansion f(x)
- of any one of a large class of functions f(x)
- by finding the values of hte coefficients an in terms of the function f(x) and its derivatives
- in Chapter 14
- we will prove the theorems that establish the validity and uniqueness of the expressions found this way