Calculus: Analytic Geometry and Calculus, with Vectors

(^586) Partial derivatives and hencef(r + hu) = f(x + hul, y + hue, z + hu3) when u = uli + u2j + uk. Then, assuming that f has ...

12 Series 12.1 Definitions and basic theorems Even though we have already had experiences with series, we start ab initio to dev ...

588 Series is not the number 5. Nevertheless, we find it very convenient to abbre- viate the statement that the series converges ...

12.1 Definitions and basic theorems 589 series is divergent. A little thought about this matter can lead us to the idea that a s ...

(^590) Series Theorem 12.15 If Euk is a series of nonnegative terms, so that uk > 0 for each k, and if the sequence sl, s2, o ...

12.1 Definitions and basic theorems 591 in which it is supposed that x is a positive number and ao, a,, a2, are numbers, not nec ...

(^592) Series It follows from Theorem 12.15 that the series 2;pk and Eqk are both con- vergent. It then follows from Theorem 12. ...

12.1 Definitions and basic theorems 593 4 Considering separately the cases in which x = 0, 0 < x < 1, and -1 < x < 0 ...

594 Series by showing that if s" is the sum of n (that is, the first n) terms of the series, then s"=(1- +( +( -4)-I- +`n n I d ...

12.1 Definitions and basic theorems 595 and hence must be convergent. Try to find simple reasons why $ <s <. If more time ...

596 Series whenever the series on the right are convergent. Solution: The inequality 0 (Iukl - I I) = IZ[kI - 21,,k1 I^xI + IVkI ...

12.1 Definitions and basic theorems 597 we suppose that e = in/n, where in and n are integers, we can suppose that n > 0 and ...

(^598) Series must converge to log 2, we enter the construction business. Let Qi, Q2, Qa, be a sequence of numbers. Let u(1), u( ...

12.2 Ratio test and integral test 599 Let e > 0. Choose an index P such that (4) Ixn - LI < e/2 (n > P) Then, when both ...

(^600) Series appropriate times. We begin by fumbling with the question whether the series (12.21) converges when x = 0.99. We s ...

12.2 Ratio test and integral test 601 etcetera. Thus the series UN + uN+1 + uN+2 + UN+3 + is dominated by the convergent series ...

602 Series way to appraise the sum of the areas of the triangular patches is to put duplicates of these patches in the rectangle ...

12.2 Ratio test and integral test 603 where 0 < C.(s) < 1. Letting n oc and using the definition (12.272) (s) _ k=1k of th ...

(^604) Series 3 Prove that if s is a constant, then the series lax + 28x2 + 38x3 + 48x4 + ... converges when lxl < 1 and dive ...

12.2 Ratio test and integral test 9 Write a complete proof of the fact that the formula x (^11199293) x2-9 _ x 1 9 =x+x3+xs+z7+ ...