**Chapter 14**

June 03 2017

**Chapter 14 The Theory of Infinite Series**
**14.1 Introduction**
- students of calculus do not always understand that infinite series are primary tools for the study of functions
- expansions of known functions have their own importance
- especially in the computation of numerical values for those functions
- however, in advanced work it often happens that an unknown power series arises from another source
- perhaps as a solution of a differential equation
- in such a case the series is used to define the otherwise unknown function which is its sum
- so the series itself is the only tool we have for investigating the properties of this function
- for many important functions of higher mathematics there is no practical alternative
**14.2 Convergent Series**
- any reasonably satisfactory study of series must be based on a careful definition of convergence for sequences
- if to each positive integer n there corresponds a definite number xn
- then the xn's are said to form a sequence
- x1, x2 ... xn ...
- we often abbreviate this array to {xn}
- it is clear that a sequence is nothing but a function defined for all positive integers n
- with the emphasis placed on the subscript notation xn instead of the function notation x(n)
- the numbers constituting sequence are called its terms
- not every sequence has a simple formula, or even any formula at all
- it is sometimes convenient to allow a sequence to start with the zeroth or second term
- a sequence {xn} is said to be bounded if there are two numbers A and B
- such that A <= xn <= B for every n
- our main interest is in the concept of the limit of a sequence
- certain sequences {xn} have the property that the numbers xn get closer and closer to some real number L as n increases
- xn = n - 1 / n
- lim n > inf n - 1 / n = 1
- a sequence {xn} is said to have a number L as a limit if for each positive number e there exists a positive integer n0 with the property that
- | xn - L | < e for all n > n0
- when L is related to {xn} in this way
- we write lim n>inf xn = L, lim xn = L
- we say that xn converges to L
- xn > L as n > inf
- or simply xn > L
- the definition requires that each e
- no matter how small
- have at least one corresponding n0 that "works" in the sense expressed
- in general, we expect that for smaller e's, larger n0's will be needed
- that is
- when the required measure of closeness is made smaller, we must go further out in the sequence to satisfy it
- a sequence is said to converge if it has a limit
- a convergent sequence is bounded
- but not all bounded sequences are convergent
- it is not always easy to decide whether a given sequence converges
- the following facts are often useful in problems of this kind
- if xn > L and yn > M then
- lim xn + yn = L + M
- lim xn - yn = L - M
- lim xn yn = L M
- lim xn / yn = L / M, if M != 0
- in working with sequences with infinite series
- we will often need to be able to recognize that a sequence is convergent
- even though we know nothing about the numerical value of hte limit
- in such a case we cannot make any direct use of the definition of a limit
- we now discuss a very important method for handling such situations
- a sequence {xn} is said to be increasing if
- x1 <= x2 <= ...
- an increasing sequence converges if and only if it is bounded
- this criterion can be used to show that in addition to
- lim n > inf (1 + 1/n)n = e
- lim n > inf (1 + 1/1! + 1/2! + ...) = e
**14.3 General Properties of Convergent Series**
- if a1, a2 ... an is a sequence of numbers
- then the expression
- E an = a1 + a2 + ...
- is called an infinite series
- or simply a series
- and the an's are called its terms
- to attach a numerical value to this expression in a natural and useful way
- we form the sequence of partial sums
- sn = a1 + a2 + ...
- this series E an is said to converge
- if the sequence {sn} converges
- if lim sn = s
- then we say that the series converges to s
- a1 + a2 + ... = s or E an = s
- the simplest and most important series is the familiar geometric series
- E xn = 1 + x + x2 + ...
- which converges to 1/(1 - x) for |x| < 1
- the most direct method for studying the convergence of a series
- is to find a closed formula for its nth partial sum
- the main disadvantage of this approach is that it rarely works
- this forces us to rely mostly on various indirect methods for establishing the convergence or divergence of series
- the main indirect method rests on the convergence criterion
- on the fact that an increasing sequence converges if and only if it is bounded
- so if the terms of our series are all nonnegative numbers
- then we clearly have
- sn <= sn + an+1 = sn+1
- so sn is an increasing sequence
- it follows that {sn} of partial sums
- and with it the series
- converges if and only if the sn's have an upper bound
- the harmonic series
- E 1/n = 1 + 1/2 + 1/3 ...
- diverges to infinity
- a great many interesting series
- some convergent and others divergent
- can be obtained from the harmonic series by thinning it out
- by deleting terms according to a systematic pattern
- if we remove all terms except reciprocals of powers of 2
- what remains is the convergent geometric series
- E 1/2n = 1 + 1/2 + 1/4 + ...
- if we remove all terms except reciprocals of primes
- the resulting series diverges
- E 1/pn = 1/2 + 1/3 + ... = inf
- the simplest general principle in deciding whether a series converges or not is the
- nth term test
- if E an converges
- then an > 0
- since an = sn - sn-1 > s - s = 0
- so an > 0 is a necessary condition for convergence
- in the sense that it follows from the convergence of the series E an
- unfortunately, it is not a sufficient condition
- for example
- the harmonic series E 1/n
- diverges even though 1/n > 0
- these examples provide a small supply of specific series of known convergence behavior
- where this behavior is decidable by rather elementary means
- the value of these familiar series for determining the behavior of new series by various methods of comparison will begin to appear next
**14.4 Series of Nonnegative Terms. Comparison Tests**
- the easiest infinite series to work with are those whose terms are all nonnegative numbers
- the reason for this
- is that the total theory of these series can be expressed by the simple statement
- if an >= 0
- then the series E an converges if and only if its sequence {sn} of partial sums is bounded
- so, in order to establish the convergence of a series of nonnegative terms
- it suffices to show that its terms approach zero fast enough to keep the potential sums bounded
- how fast is "fast enough"?
- at least as fast as the terms of a known convergent series of nonnegative terms
- the idea is contained in a formal statement called the comparison test
- if 0 <= an <= bn then
- E an converges if E bn converges
- E bn diverges if E an diverges
- we can disregard any finite number of terms at the beginning of a series if we are interested only in deciding convergence
- so the condition for the comparison test need not hold for all n
- but only for all n from some point on
- the comparison test is very simple in principle
- but in complicated cases it can be difficult to establish the necessary inequality between the nth terms of the two series being compared
- since limits are often easier to work with than inequalities
- the limit comparison test is a more convenient tool
- if E an and E bn are series with positive terms such that
- lim n > inf an / bn = 1
- then either both series converge or both series diverge
- the limit comparison test is slightly more convenient to use as
- lim n > inf an / bn = L
- where 0 < L < inf
- in using the limit comparison test we must try to guess the probable behavior of E an
- by estimating the "order of magnitude" of the nth term an
- we must try to judge whether an is approximately equal to a constant multiple of the nth term of some familiar series
- to apply this method effectively
- it is clearly desirable to have at our disposal a "stockpile" of comparison series of known behavior
- if a convergent series of nonnegative terms is rearranged in any manner
- then the resulting series also converges and has the same sum
- this isn't true if the terms are not all nonnegative
**14.5 The Integral Test. Euler's Constant**
- among the simplest infinite series are those whose terms form a decreasing sequence of positive numbers
- we study certain series of this type by means of improper integrals of the form
- S n inf f(x) dx = lim b>inf S n b f(x) dx
- the integral on the left is said to be convergent if the limit on the right exists
- in this case the value of hte integral is by definition the value of the limit
- if this limit does not exist
- then the integral is called divergent
- there is an obvious analogy to the corresponding definition for series
- E an = lim k > inf E an
- we will exploit this analogy by using integrals to obtain information about the series
- consider a series
- E an = a1 + a2 + ...
- whose terms are positive and decreasing
- in most cases the nth term an
- is a function of n given by an = f(n)
- suppose that the function y = f(x) obtained by substituting the continuous variable x in place of the discrete variable n is a decreasing function of x for x >= 1
- since f(x) is decreasing
- the rectangles of areas a1, a2 ..
- have a greater combined area than
- the area under the curve from x = 1 to x = n + 1
- so
- a1 + a2 + ... >= S 1 n+1 f(x) dx >= S 1 n f(x) dx
- similarly
- if we ignore the first rectangle with area a1
- a2 + a3 + ... <= S 1 n f(x) dx
- including a1 gives
- a1 + a2 + ... <= a1 + S 1 n f(x) dx
- by combine these inequalities
- S 1 n f(x) dx <= a1 + a2 + ... <= a1 + S 1 n f(x) dx
- this allows us to establish the integral test
- if f(x) is a positive decreasing function for x >= 1
- with the property that f(n) = an
- for each positive integer n
- then the series and integral
- E an and S 1 inf f(x) dx
- converge or diverge together
- it is clear that the integral test holds for any interval of the form x >= k
- not just for x >= 1
- we return to the inequalities
- by subtracting the integral that occurs on the left
- 0 <= a1 + a2 + .. an - S 1 n f(x) dx <= a1
- if we denote the quantity in the middle F(n)
- then 0 <= F(n) <= a1
- F(n) is a decreasing sequence
- since any decreasing sequence of nonnegative numbers converge
- the limit exists and satisfies 0 <= L <= a1
- L = lim n > inf F(n) = lim n > inf [ a1 + a2 + .. an - S 1 n f(x) dx ]
- as the main application of these ideas
- we deduce the existence of the important limit
- lim n > inf (1 + 1/2 + ... 1/n - ln n)
- which is a special case
- with an = 1/n and f(x) = 1/x
- because S 1 n dx / x = ln x ]1n = ln n
- the value of the limit is usually denoted
- y (gamma) and is called Euler's constant
- y = lim n > inf (1 + 1/2 + ... 1/n - ln n)
**14.6 The Ratio Test and Root Test**
- in the case of the geometric series
- E rn with r > 0
- the ratio a n+1 / an has constant value r
- we know this series converges if r < 1
- essentially because for these r's
- the ratio guarantees the terms decrease rapidly
- analogy leads us to expect any series E an of positive terms with converge if the ratio a n+1 / an is small for large n
- even though this ratio may not have a constant value
- these ideas are made precise in the ratio test
- if E an is a series of positive terms such that
- lim n > inf a n+1 / an = L
- then
- if L < 1, the series converges
- if L > 1, the series diverges
- if L = 1, the test is inconclusive
- the root test is especially useful for handling series whose nth term an is given by a formula that involves various products
- since a n+1 / an can often be simplified by cancellations
- we now discuss the so called root test
- which is another convenient tool for studying the convergence behavior of series
- suppose that E an is a series of nonnegative terms with the property that from some point on we have
- an <= rn, where 0 < r < 1
- the geometric series E rn clearly converges
- so E an also converges by the comparison test
- the fact that the inequalities can be written in the form
- n sqrt (an) <= r < 1
- brings us to a convenient statement of the root test
- if E an is a series of nonnegative terms such that lim n > inf n sqrt (an) = L
- then
- if L < 1, the series converges
- if L > 1, the series diverges
- if L = 1, the series is inconclusive
- in general
- the root test is most likely to be useful for treating series in which an is complicated but n sqrt (an) is simple
**14.7 The Alternating Series Test. Absolute Convergence**
- we now consider series with both positive and negative terms
- the simplest are those whose terms are alternatively positive and negative
- these are called alternating series
- E (-1)n+1 an = a1 - a2 + a3 ...
- as examples we have already seen
- 1 - 1/2 + 1/3 - 1/4 + ... = ln 2
- 1 - 1/3 + 1/5 - 1/7 + ... = pi / 4
- it is easy to see that both of the alternating series have th property that the an's form a decreasing sequence that approaches zero
- a1 >= a2 >= ...
- an > 0
- Leibniz noticed these two simple conditions are enough to guarantee that any alternating series converges
- this is the alternating series test
- the partial sums is similar to a swinging pendulum
- oscillating back and forth
- slowly approaching an equilibrium position
- some series with terms of mixed signs do not need the assistance of minus signs for convergence
- but converge because of the smallness of their terms alone
- they would still converge even if all the minus signs were replaced by plus signs
- a series E an is absolutely convergent if E |an| converges
- absolute convergence is a stronger property than ordinary convergence
- absolute converge implies convergence
- when trying to establish convergence of a series where terms have mixed signs
- testing for absolute convergence is a good first step
- all our previous tests
- the comparison tests, integral test, ratio test, and root test
- apply only to series of positive terms
**14.8 Power Series Revisited. Interval of Convergence**
- in the preceding sections we concentrated our attention on series whose terms are constants
- in the next five sections we turn to the study of power series
- whose terms are very simple functions of variable x
- we now approach power series from a different point of view
- power series is a series of the form
- E an xn = a0 + a1x + a2x2 + ...
- a differential equation can often be used to generate a solution of itself in the form of a power series
- it is perfectly reasonable to define a function f(x) by saying it is the sum of this power series
- f(x) = E an xn
- provided that the series converges
- the geometric series
- E xn = 1 + x + x2 + ...
- is the simplest power series
- we know this series converges for |x| < 1 and diverges for |x| >= 1
- in general
- we expect a power series to converge for some values of x and diverge for others
- we will be very interested in knowing the x's for which a given power series E an xn converges
- for any such x
- the sum of the series is a number whose value depends on x
- and is therefore a function of x
- if we denote this function f(x)
- then f(x) can be thought of as defined by
- f(x) = E an xn
- sometimes a power series has a known function as its sum
- for example
- if |x| < 1
- we know the geometric series has 1 / 1-x as its sum
- however
- in general there is no reason to expect the sum of a convergent power series will turn out to be a function we recognize from previous experience
- we will consider two major groups of questions
- what properties does the function f(x) defined by E an xn have
- if a function f(x) is given beforehand, under what circumstances does it have a power series expansion of the form E an xn
- our task is to discover the structure of the set of all x's for which a given power series converges
- there are only three possibilities
- the series converges only for x = 0
- the series is absolutely convergent for all x
- there exists a positive real number R such that the series is convergent for |x| < R and divergent for |x| > R
- every power series has a radius of convergence R
- where 0 <= R <= inf
- with the property that the series converges absolutely if |x| < R and diverges if |x| > R
- there is a simple formula from the ratio test for R that works in many situations
- R = lim | an / an+1 |
- provided this limit exists
- and has inf as an allowed value
- the second step is to test the behavior of the series at the endpoints
**14.9 Differentiation and Integration of Power Series**
- consider a power series E an xn with a positive radius of convergence R
- this series can be used to define a function f(x) whose domain of definition is the interval of convergence of the series
- for each x in this interval we define f(x) to be the sum of the series
- f(x) = a0 + a1x + a2x2 + ...
- we say E an xn is a power series expansion of f(x)
- for example
- if |x| < 1
- then 1 / 1+x = 1 - x + x2 - ...
- because the geometric series E (-1)n xn converges and has the sum 1/1+x
- polynomials are finite sums of terms of the form an xn, are very simple functions
- they are continuous everywhere
- and can be differentiated and integrated term by term
- the sum of a power series can be a much more complicated function
- but it is still simple enough to share three properties with polynomials inside the interval of convergence
- f(x) is continuous on the open interval (-R, R)
- f(x) is differentiable on (-R, R), the derivative is f'(x) = a1 + 2a2x + ...
- if x is any point on (-R, R), then S0x f(t) dt = a0x + 1/2 a1 x2 + ...
- in the interior of its interval of convergence
- a power series defines an infinitely differentiable function whose derivatives can be calculated by differentiating the series term by term
- d/dx E ax xn = E d/dx an xn
- term-by-term differentiability of a convergent series of functions is usually false
- we can avoid the dummy variable t by writing
- S f(x) dx = a0 x + 1/2 a1 x2 + ...
- provided we find an indefinite integral on the left that equals zero when x = 0
- the term-by-term integration of a power series can be emphasized by
- S E ax xn = E S an xn dx
- the differentiated and integrated series converge on the interval (-R, R)
**14.10 Taylor Series and Taylor's Formula**
- we have solved the problem of determining the general nature of the sum of a power series
- inside the interval of convergence
- it is a continuous function with derivatives of all orders
- we now investigate the converse problem of starting with a given infinitely differentiable function and expanding it in a power series
- in section 14.9 we established several such expansions for a few special functions with particularly simple derivatives
- we now consider a method of much more generality
- it may seem the coefficients of a power series are not connected with one another in any necessary way
- in fact, they are bound together by an invisible chain
- which we now make visible
- assume f(x) is the sum of a power series with positive radius of convergence
- f(x) = E an xn , R > 0
- in general
- f(n) (x) = n!an + terms continuing x as a factor
- we know these series expansions of the derivatives are valid on teh open interval |x| < R
- by putting x = 0
- f(n) (0) = n!an
- so an = f(n) (0) / n!
- f(x) = f(0) + f'(0) x + ... + f(n) (0)/n! xn + ...
- is called the Taylor series of f(x) at x = 0
- if a function is represented by a power series with positive radius of convergence
- then there is only one such series and it must be the Taylor series of the function
- power series are unique
- because the coefficients are uniquely determined by the function itself
- f(x) = E f(n) (0)/n! xn
- the numbers an = f(n) (0) / n! are called the Taylor coefficients of f(x)
- the equation
- f(x) = f(0) + f'(0) x + ... + f(n) (0)/n! xn + ...
- is true because we started with a convergent series having f(x) as its sum
- we now start with a function f(x) that has derivatives of all orders throughout some open interval I containing the point x = 0
- we can form the Taylor series on the right
- and ask is the Taylor series a valid expansion of f(x) on the interval I
- the equation is not always valid
- whether it is or not depends entirely on the individual nature of function f(x)
- to show a Taylor series expansion of f(x) is valid
- we must show the remainder of the series Rn(x) after xn approaches 0
- the Taylor series expansions of ex, sin x, and cos x are valid
- and can be used to calculate indefinite integrals that cannot be calculated in terms of elementary functions
- such as S01 e -x2 dx