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