Calculus: Analytic Geometry and Calculus, with Vectors

606 Series and show that n-1 (2) j o n2 + V >Jonn2 + x2dx = n tan 1 n]o Use this result to show that n-1 (3) 2 2=. n=1 k=o n ...

12.2 Ratio test and integral test 607 17 With the aid of an appropriate figure show that, whens > 3, 3+4,+... +nd< (- Idx= ...

(^608) Series and hence 0 is quite close to 1 even when n = 1. The above formula for n! is a Stirling formula, and it is very us ...

12.2 Ratio test and integral test 609 by choosing a number r for which 1 < r < p and showing that there is a con- stant 14 ...

(^610) Series then existence of 1 aW lg(x)l dx implies existence of fag f(x) dx. Solution: The theorem of Problem 20 implies tha ...

12.3 Alternating series and Fourier series 611 so on. Because the quantities in parentheses are positive, the formulas (12.314) ...

612 E and is such that the two integrals (12.34) fE f(x) dx, fE if(x)I' dx Series both exist as Riemann integrals or as Cauchy e ...

12.3 Alternating series and Fourier series 613 Such sets are said to be complete. While proofs of such things are so long and de ...

614 Series when x is not an integer and [x] denotes the greatest integer less thanor equal to x. Problem 8 at the end of this se ...

12.3 Alternating series and Fourier series 615 2 Find, correct to four decimal places, the numbers to which the following series ...

(^616) Series 6 Supposing that x = r/4, give one or more reasons why the series sin x sin 2x sin 3x sin 4x 1 2 3 4 + is not an a ...

12.3 Alternating series and Fourier series 617 Then tell why (12.37) must be valid and substitute in it to obtain ax 4 ax 1 3- I ...

(^618) Series synthesis. Let (ki, 02, 03, .. - be a set of functions orthonormal over a set E. Let a,, a2, a3, be given coeffici ...

12.4 Power series 619 is'. alid Ns henever the series on the right is convergent. From this it follows that (5) limF(x - t) - 2F ...

620 converge for each x. The geometric series in the formula (12.415) 1 = 1 +x+x2+x3+ 1 - x Series is an example of a power seri ...

12.4 Power series 621 that is, (12.45) f." a x cl(t - a) dt + J c_(t -a)2dt f. z co dt -} a a + J c3(t - a)3 dt +. a = co(x - a) ...

622 Series For example, the series in (12.412), (12.413), and (12.414) are the only power series in x that converge to ex, cos x ...

12.4 Power series 623 and put x = 0 to find c2. Continue the process until c5 has been obtained. Finally, see whether the result ...

624 Series converge to f(x) and g(x) when jxj < r, then the series in (3) will converge to f(x)g(x) when ixi < r provided ...

12.4 Power series 625 Use this idea and the known power series expansion of cosx to obtain some of the coefficients in the expan ...