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