Comments
Description
Transcript
Summation of series
SERIES AND LIMITS some sort of relationship between successive terms. For example, if the nth term of a series is given by un = 1 , 2n for n = 1, 2, 3, . . . , N then the sum of the first N terms will be SN = N n=1 un = 1 1 1 1 + + + ··· + N. 2 4 8 2 (4.1) It is clear that the sum of a finite number of terms is always finite, provided that each term is itself finite. It is often of practical interest, however, to consider the sum of a series with an infinite number of finite terms. The sum of an infinite number of terms is best defined by first considering the partial sum of the first N terms, SN . If the value of the partial sum SN tends to a finite limit, S, as N tends to infinity, then the series is said to converge and its sum is given by the limit S. In other words, the sum of an infinite series is given by S = lim SN , N→∞ provided the limit exists. For complex infinite series, if SN approaches a limit S = X + iY as N → ∞, this means that XN → X and YN → Y separately, i.e. the real and imaginary parts of the series are each convergent series with sums X and Y respectively. However, not all infinite series have finite sums. As N → ∞, the value of the partial sum SN may diverge: it may approach +∞ or −∞, or oscillate finitely or infinitely. Moreover, for a series where each term depends on some variable, its convergence can depend on the value assumed by the variable. Whether an infinite series converges, diverges or oscillates has important implications when describing physical systems. Methods for determining whether a series converges are discussed in section 4.3. 4.2 Summation of series It is often necessary to find the sum of a finite series or a convergent infinite series. We now describe arithmetic, geometric and arithmetico-geometric series, which are particularly common and for which the sums are easily found. Other methods that can sometimes be used to sum more complicated series are discussed below. 116 4.2 SUMMATION OF SERIES 4.2.1 Arithmetic series An arithmetic series has the characteristic that the difference between successive terms is constant. The sum of a general arithmetic series is written SN = a + (a + d) + (a + 2d) + · · · + [a + (N − 1)d] = N−1 (a + nd). n=0 Rewriting the series in the opposite order and adding this term by term to the original expression for SN , we find SN = N N [a + a + (N − 1)d] = (first term + last term). 2 2 (4.2) If an infinite number of such terms are added the series will increase (or decrease) indefinitely; that is to say, it diverges. Sum the integers between 1 and 1000 inclusive. This is an arithmetic series with a = 1, d = 1 and N = 1000. Therefore, using (4.2) we find 1000 (1 + 1000) = 500500, 2 which can be checked directly only with considerable effort. SN = 4.2.2 Geometric series Equation (4.1) is a particular example of a geometric series, which has the characteristic that the ratio of successive terms is a constant (one-half in this case). The sum of a geometric series is in general written SN = a + ar + ar 2 + · · · + ar N−1 = N−1 ar n , n=0 where a is a constant and r is the ratio of successive terms, the common ratio. The sum may be evaluated by considering SN and rSN : SN = a + ar + ar 2 + ar 3 + · · · + ar N−1 , rSN = ar + ar 2 + ar 3 + ar 4 + · · · + ar N . If we now subtract the second equation from the first we obtain (1 − r)SN = a − ar N , and hence SN = a(1 − r N ) . 1−r 117 (4.3) SERIES AND LIMITS For a series with an infinite number of terms and |r| < 1, we have limN→∞ r N = 0, and the sum tends to the limit a S= . (4.4) 1−r In (4.1), r = 12 , a = 12 , and so S = 1. For |r| ≥ 1, however, the series either diverges or oscillates. Consider a ball that drops from a height of 27 m and on each bounce retains only a third of its kinetic energy; thus after one bounce it will return to a height of 9 m, after two bounces to 3 m, and so on. Find the total distance travelled between the first bounce and the Mth bounce. The total distance travelled between the first bounce and the Mth bounce is given by the sum of M − 1 terms: M−2 9 SM−1 = 2 (9 + 3 + 1 + · · · ) = 2 3m m=0 for M > 1, where the factor 2 is included to allow for both the upward and the downward journey. Inside the parentheses we clearly have a geometric series with first term 9 and common ratio 1/3 and hence the distance is given by (4.3), i.e. M−1 9 1 − 13 M−1 , = 27 1 − 13 SM−1 = 2 × 1 1− 3 where the number of terms N in (4.3) has been replaced by M − 1. 4.2.3 Arithmetico-geometric series An arithmetico-geometric series, as its name suggests, is a combined arithmetic and geometric series. It has the general form SN = a + (a + d)r + (a + 2d)r 2 + · · · + [a + (N − 1)d] r N−1 = N−1 (a + nd)r n , n=0 and can be summed, in a similar way to a pure geometric series, by multiplying by r and subtracting the result from the original series to obtain (1 − r)SN = a + rd + r 2 d + · · · + r N−1 d − [a + (N − 1)d] r N . Using the expression for the sum of a geometric series (4.3) and rearranging, we find rd(1 − r N−1 ) a − [a + (N − 1)d] r N + SN = . 1−r (1 − r)2 For an infinite series with |r| < 1, limN→∞ r N = 0 as in the previous subsection, and the sum tends to the limit rd a + . (4.5) S= 1 − r (1 − r)2 As for a geometric series, if |r| ≥ 1 then the series either diverges or oscillates. 118 4.2 SUMMATION OF SERIES Sum the series S =2+ 5 11 8 + 3 + ··· . + 2 22 2 This is an infinite arithmetico-geometric series with a = 2, d = 3 and r = 1/2. Therefore, from (4.5), we obtain S = 10. 4.2.4 The difference method The difference method is sometimes useful in summing series that are more complicated than the examples discussed above. Let us consider the general series N un = u1 + u2 + · · · + uN . n=1 If the terms of the series, un , can be expressed in the form un = f(n) − f(n − 1) for some function f(n) then its (partial) sum is given by SN = N un = f(N) − f(0). n=1 This can be shown as follows. The sum is given by SN = u1 + u2 + · · · + uN and since un = f(n) − f(n − 1), it may be rewritten SN = [ f(1) − f(0)] + [ f(2) − f(1)] + · · · + [ f(N) − f(N − 1)]. By cancelling terms we see that SN = f(N) − f(0). Evaluate the sum N n=1 Using partial fractions we find 1 . n(n + 1) un = − 1 1 − n+1 n . Hence un = f(n) − f(n − 1) with f(n) = −1/(n + 1), and so the sum is given by SN = f(N) − f(0) = − 1 N +1= . N+1 N+1 119 SERIES AND LIMITS The difference method may be easily extended to evaluate sums in which each term can be expressed in the form un = f(n) − f(n − m), (4.6) where m is an integer. By writing out the sum to N terms with each term expressed in this form, and cancelling terms in pairs as before, we find SN = m f(N − k + 1) − k=1 m f(1 − k). k=1 Evaluate the sum N n=1 Using partial fractions we find un = − 1 . n(n + 2) 1 1 . − 2(n + 2) 2n Hence un = f(n) − f(n − 2) with f(n) = −1/[2(n + 2)], and so the sum is given by 3 1 1 1 . + SN = f(N) + f(N − 1) − f(0) − f(−1) = − 4 2 N+2 N+1 In fact the difference method is quite flexible and may be used to evaluate sums even when each term cannot be expressed as in (4.6). The method still relies, however, on being able to write un in terms of a single function such that most terms in the sum cancel, leaving only a few terms at the beginning and the end. This is best illustrated by an example. Evaluate the sum N n=1 1 . n(n + 1)(n + 2) Using partial fractions we find un = 1 1 1 − + . 2(n + 2) n + 1 2n Hence un = f(n) − 2f(n − 1) + f(n − 2) with f(n) = 1/[2(n + 2)]. If we write out the sum, expressing each term un in this form, we find that most terms cancel and the sum is given by 1 1 1 1 . SN = f(N) − f(N − 1) − f(0) + f(−1) = + − 4 2 N+2 N+1 120 4.2 SUMMATION OF SERIES 4.2.5 Series involving natural numbers Series consisting of the natural numbers 1, 2, 3, . . . , or the square or cube of these numbers, occur frequently and deserve a special mention. Let us first consider the sum of the first N natural numbers, SN = 1 + 2 + 3 + · · · + N = N n. n=1 This is clearly an arithmetic series with first term a = 1 and common difference d = 1. Therefore, from (4.2), SN = 12 N(N + 1). Next, we consider the sum of the squares of the first N natural numbers: SN = 12 + 22 + 32 + . . . + N 2 = N n2 , n=1 which may be evaluated using the difference method. The nth term in the series is un = n2 , which we need to express in the form f(n) − f(n − 1) for some function f(n). Consider the function f(n) = n(n + 1)(2n + 1) ⇒ f(n − 1) = (n − 1)n(2n − 1). For this function f(n) − f(n − 1) = 6n2 , and so we can write un = 16 [ f(n) − f(n − 1)]. Therefore, by the difference method, SN = 16 [ f(N) − f(0)] = 16 N(N + 1)(2N + 1). Finally, we calculate the sum of the cubes of the first N natural numbers, SN = 13 + 23 + 33 + · · · + N 3 = N n3 , n=1 again using the difference method. Consider the function f(n) = [n(n + 1)]2 ⇒ f(n − 1) = [(n − 1)n]2 , for which f(n) − f(n − 1) = 4n3 . Therefore we can write the general nth term of the series as un = 14 [ f(n) − f(n − 1)], and using the difference method we find SN = 14 [ f(N) − f(0)] = 14 N 2 (N + 1)2 . Note that this is the square of the sum of the natural numbers, i.e. N 2 N 3 n = n . n=1 n=1 121 SERIES AND LIMITS Sum the series N (n + 1)(n + 3). n=1 The nth term in this series is un = (n + 1)(n + 3) = n2 + 4n + 3, and therefore we can write N (n + 1)(n + 3) = N n=1 (n2 + 4n + 3) n=1 = N n=1 n2 + 4 N n=1 n+ N 3 n=1 = 16 N(N + 1)(2N + 1) + 4 × 12 N(N + 1) + 3N = 16 N(2N 2 + 15N + 31). 4.2.6 Transformation of series A complicated series may sometimes be summed by transforming it into a familiar series for which we already know the sum, perhaps a geometric series or the Maclaurin expansion of a simple function (see subsection 4.6.3). Various techniques are useful, and deciding which one to use in any given case is a matter of experience. We now discuss a few of the more common methods. The differentiation or integration of a series is often useful in transforming an apparently intractable series into a more familiar one. If we wish to differentiate or integrate a series that already depends on some variable then we may do so in a straightforward manner. Sum the series S(x) = x4 x5 x6 + + + ··· . 3(0!) 4(1!) 5(2!) Dividing both sides by x we obtain x3 x4 x5 S(x) = + + + ··· , x 3(0!) 4(1!) 5(2!) which is easily differentiated to give x2 x3 x4 x5 d S(x) = + + + + ··· . dx x 0! 1! 2! 3! Recalling the Maclaurin expansion of exp x given in subsection 4.6.3, we recognise that the RHS is equal to x2 exp x. Having done so, we can now integrate both sides to obtain S(x)/x = x2 exp x dx. 122 4.2 SUMMATION OF SERIES Integrating the RHS by parts we find S(x)/x = x2 exp x − 2x exp x + 2 exp x + c, where the value of the constant of integration c can be fixed by the requirement that S(x)/x = 0 at x = 0. Thus we find that c = −2 and that the sum is given by S(x) = x3 exp x − 2x2 exp x + 2x exp x − 2x. Often, however, we require the sum of a series that does not depend on a variable. In this case, in order that we may differentiate or integrate the series, we define a function of some variable x such that the value of this function is equal to the sum of the series for some particular value of x (usually at x = 1). Sum the series S =1+ 2 4 3 + 3 + ··· . + 2 22 2 Let us begin by defining the function f(x) = 1 + 2x + 3x2 + 4x3 + · · · , so that the sum S = f(1/2). Integrating this function we obtain f(x) dx = x + x2 + x3 + · · · , which we recognise as an infinite geometric series with first term a = x and common ratio r = x. Therefore, from (4.4), we find that the sum of this series is x/(1 − x). In other words x f(x) dx = , 1−x so that f(x) is given by f(x) = 1 d x = . dx 1 − x (1 − x)2 The sum of the original series is therefore S = f(1/2) = 4. Aside from differentiation and integration, an appropriate substitution can sometimes transform a series into a more familiar form. In particular, series with terms that contain trigonometric functions can often be summed by the use of complex exponentials. Sum the series S(θ) = 1 + cos θ + cos 2θ cos 3θ + + ··· . 2! 3! Replacing the cosine terms with a complex exponential, we obtain exp 2iθ exp 3iθ S(θ) = Re 1 + exp iθ + + + ··· 2! 3! (exp iθ)2 (exp iθ)3 + + ··· . = Re 1 + exp iθ + 2! 3! 123