Advanced High-School Mathematics

SECTION 5.2 Numerical Series 271 nlim→∞ ( √^1 n+ 1 ) ( 1 n ) =∞, showing that the terms of the series ∑∞ n=0 1 √ n+ 1 are asympt ...

272 CHAPTER 5 Series and Differential Equations nlim→∞ 1 (ln(n)^2 1 n ) = nlim→∞ n (lnn)^2 = xlim→∞ x (lnx)^2 l’Hˆopital= lim x→ ...

SECTION 5.2 Numerical Series 273 the same behavior as the improper integral ∫∞ 1 dx xp . But, wherep 6 = 1, we have ∫∞ 1 dx xp = ...

274 CHAPTER 5 Series and Differential Equations seriesa+ar+ar^2 +··· converges (to a 1 −r ). In this test we do not need to assu ...

SECTION 5.2 Numerical Series 275 nlim→∞ an+1 an = limn→∞ Å(n+2) 3 (n+1)! ã Å(n+1) 3 n! ã = limn→∞ (n+ 2)^3 (n+ 1)(n+ 1)^3 = 0. T ...

276 CHAPTER 5 Series and Differential Equations (c) ∑∞ n=1 n^2 √ n^7 + 2n (d) ∑∞ n=0 n^2 + 2n 3 n (e) ∑∞ n=0 (n+ 1)3n n! (f) ∑∞ ...

SECTION 5.2 Numerical Series 277 As in the above sequence, LetF 0 , F 1 , F 2 , ... be the terms of the Fibonacci sequence. Sho ...

278 CHAPTER 5 Series and Differential Equations We consider a couple of simple illustrations of the above theorem. Example 1.The ...

SECTION 5.2 Numerical Series 279 series (do this!). However, the terms decrease and tend to zero and so by theAlternating Series ...

280 CHAPTER 5 Series and Differential Equations (See the footnote.^16 ) By thinking in terms of binary decimal representations ( ...

SECTION 5.2 Numerical Series 281 ∑n k=1 akbk = snbn+1+ ∑n k=1 sk(bk−bk+1). Proof. Settings 0 = 0 we obviously haveak=sn−sk− 1 , ...

282 CHAPTER 5 Series and Differential Equations Sincebk → 0 ask → ∞, we conclude that (rn) is a Cauchy sequence of real numbers, ...

SECTION 5.3 Concept of Power Series 283 a+ar+ar^2 +··· = a 1 −r . Let’s make a minor cosmetic change: rather than writingr in th ...

284 CHAPTER 5 Series and Differential Equations 5.3.1 Radius and interval of convergence Our primary tool in determining the con ...

SECTION 5.3 Concept of Power Series 285 R = limn→∞ ∣∣ ∣∣ ∣ an an+1 ∣∣ ∣∣ ∣ = limn→∞ (n 2 n ) (n+1 2 n+1 ) = limn→∞^2 n n+ 1 = 2, ...

286 CHAPTER 5 Series and Differential Equations In the examples above we computed intervals within which we are guaranteedconver ...

SECTION 5.3 Concept of Power Series 287 ∑∞ n=0 n 2 n 2 n = ∑∞ n=0 n which also diverges. Therefore, ∑∞ n=0 nxn 2 n has interval ...

288 CHAPTER 5 Series and Differential Equations Determine the interval convergence of each of the power series below: (a) ∑∞ n ...

SECTION 5.4 Polynomial Approximations 289 Notice that as long asxdoes not move too far away from the pointa, then the above appr ...

290 CHAPTER 5 Series and Differential Equations Such a quadratic function is actually very easy to build: the result would be th ...