Advanced High-School Mathematics

SECTION 5.1 Quick Survey of Limits 251 Next, one knows that ∑n i=1 i^3 =^14 n^2 (n+ 1)^2 ; therefore, U(f;P) = 4 n^2 (n+ 1)^2 n^ ...

252 CHAPTER 5 Series and Differential Equations (b) limx→af(x) = L, (c) gis defined in a punctured neighborhood ofL, and (d) lim ...

SECTION 5.1 Quick Survey of Limits 253 Show that LUB(A) = 1 and GLB(A) = 0.^4 (The irrationality of π.) This exercise will guid ...

254 CHAPTER 5 Series and Differential Equations = ∑n i=0 (−1)if(2i)(x), n≥ 1 , and show that (e) Fn(0) andFn(π) are both integer ...

SECTION 5.1 Quick Survey of Limits 255 The definitions of these improper integrals are in terms of limits. For example ∫∞ 0 f(x) ...

256 CHAPTER 5 Series and Differential Equations This is a fairly simple integration: using the substitution (u= lnx) one first c ...

SECTION 5.1 Quick Survey of Limits 257 Now letf be a continuous function defined for all real numbers and compute Tlim→ 0 1 T ∫∞ ...

258 CHAPTER 5 Series and Differential Equations At the same time, you’ll no doubt remem- ber that the computation of this limit ...

SECTION 5.1 Quick Survey of Limits 259 This result we summarize as l’Hˆopital’s Rule. Let f and g be functions differentiable on ...

260 CHAPTER 5 Series and Differential Equations Other indeterminate forms can be treated as in the following exam- ples. Example ...

SECTION 5.1 Quick Survey of Limits 261 Exercises Using l’Hˆopital’s rule if necessary, compute the limits indicated below: (a) ...

262 CHAPTER 5 Series and Differential Equations provided the improper integral exists. Assuming that f ∗g(x) exists for allx, sh ...

SECTION 5.1 Quick Survey of Limits 263 An important theorem of electrical engineering is that if the input voltage is x(t), then ...

264 CHAPTER 5 Series and Differential Equations (a) IfG(x) = ∫x 0 f(t)dt, thenF ′(0) =G′(0). (b) Show thatGis anoddfunction, i.e ...

SECTION 5.2 Numerical Series 265 Series 2: 1 + 1 22 + 1 32 + 1 42 +···. To see how the above two series differ, we shall conside ...

266 CHAPTER 5 Series and Differential Equations then the sequenceconvergesto some limitL(which we might not be able to compute!) ...

SECTION 5.2 Numerical Series 267 Fact 2: If Fact 1 is true, we still need to show that there is some numberM such that ∑k n=0 an ...

268 CHAPTER 5 Series and Differential Equations (a) ∑∞ n=0 1 4 n^2 − 1 (b) ∑∞ n=0 3 9 n^2 − 3 n− 2 Consider the series Σ = ∑{ ...

SECTION 5.2 Numerical Series 269 Prove that the limit limn→∞ Ñ n ∑ k=1 1 k −lnn é exists; its limit is called Euler’s constant ...

270 CHAPTER 5 Series and Differential Equations asymptotically the series ∑∞ n=0 bnis no larger thanRtimes the con- vergent seri ...