Using Difference Equations to Generalize Results for Periodic Nested Radicals

Usig Diffeece Equatios to Geealize Results fo Peiodic Nested Radicals Chis Lyd Uivesity of Rhode Islad, Depatmet of Mathematics South Kigsto, Rhode Islad

2 2 2 2 2 2 2 π = + + +... Vieta (593) 2 2 2 = + Eule (775) e 2 2+ 2 3 + 3 4 +... 3= + 2 + 3 + 4 +... Ramauja (9) 3 4 5 xe 2 = x x x x... Fieldig (2002)

Defiitio Let Let = c ad a be a sequece of iteges geate tha o equal to two. = = be sequeces of eal umbes. If the sequece = 2 2 + 2 + = 2 + + z a, a c a,..., a c a... c a,... is eal fo all Z +, the the expessio a + c 2a + c 3a + c 4a + 2 2 3 3 4... is called a ight ested adical, ad it deotes the limit of z exists., if it

Defiitio If the sequece 2 2 2 2 2 y = a, a + c a,..., a +... + c a + c a,... = is eal fo all Z +, the the expessio 4 3 2 4 4 3 3 2 2... + c a + c a + c a + c a is called a left ested adical ad it deotes the limit of exists. y, if it

Defiitio If the sequece 2 2 2 2 2 y a, a c a,..., a... c a c a,... = + + + + = is eal fo all Z +, the the expessio 4 3 2 4 4 3 3 2 2... + c a + c a + c a + c a is called a left ested adical ad it deotes the limit of exists. y, if it 2 2 If thee is a Z +, such that a +... + c 2a + c a R, the we say the sequece y is udefied. I this case, the ested adical is also udefied.

Liteatue o the topic of ested adicals []-[8] has focused o two types of mathematical poblems. Poblem Type #: Give a paticula fom of ested adical, detemie the ecessay ad sufficiet coditios fo the ested adical to covege. Fo example, Let ad 2 be iteges geate tha o equal to two. Let a ad b be iteges. Detemie the age of paametes a, b,, ad 2, fo which the ested adical 2 2 a b a b a... has a coespodig sequece that is eal ad coveges.

Liteatue o the topic of ested adicals []-[8] has focused o two types of mathematical poblems. Poblem Type #2: Detemie the set of umbes that ca be expessed as the limit of a ested adical of a give fom. Fo example, Ca evey positive itege be expessed as the limit of a ested adical of the fom 3 4 3 4... a b+ c+ a b+ c+ whee a, b, ad c ae positive iteges?

Thoughout this pesetatio, we will coside ested adicals of the fom i Defiitio whee, c, ad a ae peiodic sequeces. = = =

Thoughout this pesetatio, we will coside ested adicals of the fom i Defiitio whee, c, ad a ae peiodic sequeces. = = = We utilize thee theoems fom the field of diffeece equatios that apply to sequeces that ae geeated by iteatig cotiuous fuctios. This method of aalysis allows us to get esults i a moe geeal settig. Fo example, each of the idetities o the ext slide ca be pove usig the techiques that ae used to pove the theoems i this pesetatio.

3 4 3 4 2= 6+ 7 3 6+ 7 3... 3 3 =... + 6+ 24+ 36+ 24+ 36 2= log 97+ log 998+ log 97+ log 998 +... * 0.837... = si 9+ si 9+ si 9+ si 9 +... 4= 3+ 3= + 6 + 6 +... 3+ 2 2 3+ 2 3 +...

Theoem Suppose that the sequece, c = z =, ad a = ae peiodic sequeces such that =, coespodig to the ested adical a + c 2a + c 3a + c 4a + 2 2 3 3 4... as pe Defiitio, is a sequece of eal umbes. The the followig statemets ae tue.

Theoem Suppose that the sequece, c = z =, ad a = ae peiodic sequeces such that =, coespodig to the ested adical a + c 2a + c 3a + c 4a + 2 2 3 3 4... as pe Defiitio, is a sequece of eal umbes. (a) Thee exists a positive itege k such that z, z,...,ad z k+ = 0 k+ 2 = 0 k+ k = 0 ae solutios of the diffeece equatio 2... k + 2 k k k x = a + c a + + c a + c x fo =, 2, 3, (.) with espective iitial tems x = z = a, ad fo j = 2,, k, j j x = z = a +... + c j a. j

Theoem Suppose that the sequece, c = z =, ad a = ae peiodic sequeces such that =, coespodig to the ested adical a + c 2a + c 3a + c 4a + 2 2 3 3 4... as pe Defiitio, is a sequece of eal umbes. (b) Let z i be the fist o-zeo tem i z. The fuctio 2 2 k k k f ( x) = a + c a +... + c k a + c x (.2) is defied ad cotiuous o a closed, but ot ecessaily bouded, iteval D that eithe cotais 0, z i o z,0 i depedig upo the sig of z i.

Theoem Suppose that the sequece, c = z =, ad a = ae peiodic sequeces such that =, coespodig to the ested adical a + c 2a + c 3a + c 4a + 2 2 3 3 4... as pe Defiitio, is a sequece of eal umbes. (c) If k c < 0, the the fuctio i (.2) is deceasig o D. j= j (d) If k c > 0, the the fuctio i (.2) is iceasig o D. j= j

Theoem Suppose that the sequece, c = z =, ad a = ae peiodic sequeces such that =, coespodig to the ested adical a + c 2a + c 3a + c 4a + 2 2 3 3 4... as pe Defiitio, is a sequece of eal umbes. (e) If the sequece z coveges to a limit L, the L is a equilibium poit of the diffeece equatio i (.). (f) Suppose that the sequece z coveges to a limit L R ad suppose that c ad a ae atioal sequeces. The L is ot a tascedetal umbe.

Example: Let z be the sequece coespodig to 3 98 3 99 3 98 3 99 00 00 00 00... By Theoem, the coespodig diffeece equatio is: x 3 98 3 99 x + = This diffeece equatio has 5 equilibium poits. Fo ay solutio x, we have: So, x x x x, 2,, 4 x x x x x 2 4 3 x x x x z x ad z x 2+ = 0 5 2+ 2 = 0 5 x 3 98 3 99 2 00 00 = z = x x 2 00 00 f ( x) x 4 = x x 3 3 98 00 = z = 3 98 3 99 00 00 x 5 Thus, 3 98 3 99 3 98 3 99 00 00 00 00... does ot covege. x

Example: 3 3 3 3 3 3 3 3 2 4 8 + 64 + 2 4 8 + 64 +... coveges. 3 3 3 3 3 3 3 3 2 4 8 64 2 4 8 64... coveges to a miimal peiod-2 sequece. 3 3 3 3 3 3 3 3 2 4 + 8 64 + 2 4 + 8 64 +... coveges to a miimal peiod-4 sequece. 3 3 3 3 3 3 3 3 2 4 8 64 + 2 4 8 64 +... coveges to a miimal peiod-8 sequece.

Theoem 2 Let be a peiodic sequece of iteges geate tha o equal to two. Let c ad a be peiodic sequeces of o-egative eal umbes. The, the sequece z =, coespodig to the ested adical coveges. a 2 3 4 5 c a 2 c 2 a 3 c 3 a 4 c 4 a 5... + + + + +,

Theoem 3 Let Let = c = be a peiodic sequece of iteges geate tha o equal to two. be a peiodic sequece whee each,. Let L be ay itege geate tha o equal to two. c The, we ca costuct a peiodic sequece of positive iteges that 2 3 4 2 2 3 3 4... a + c a + c a + c a + = L. a such

Suppose we choose = 3,5,7,3,5,7,3,.... c Suppose we choose =,,,,,.... Suppose we choose L = 4. a will have a miimal peiod of 6 sice that is the least commo multiple of the peiods of ad c. We costuct = 68, 020, 6388, 60, 028, 6380, 68, 020, 6388,... a 3 ad 5 7 3 5 7 3 4= 68 020+ 6,388 60+ 028 6,380+ 68...

Theoem 4 Let p be ay positive itege. () The, oe ca costuct peiod-p sequeces of iteges, c, ad a ested adical such that the sequece z a + c 2a + c 3a + c 4a + 2 2 3 3 4..., coespodig to the ight coveges asymptotically to a sequece of miimal peiod p. (2) Also, oe ca costuct peiod-p sequeces of iteges, c, ad a adical such that the sequece z 4 3 2 4 4 3 3 2 2... + c a + c a + c a + c a, coespodig to the left ested coveges asymptotically to a sequece of miimal peiod p.

Refeeces: Ramauja, Joual of the Idia Mathematical Society, (9) p. 90 poblem #289 2 A. Heschfeld, O ifiite adicals, Ame. Math. Mothly, 42 (935) 49 429 3 W. S. Size, Cotiued oots, Math. Magazie, 59 (986) 23 27 4 S. Zimmema ad C. Ho, O ifiitely ested adicals, Mathematics Magazie, 8 (2008) 3 5 5 L. D. Sevi, Nested squae oots of 2, Ame. Math. Mothly, 0 (2003) 326 330 6 M. A. Nyblom, Moe ested squae oots of 2, Ame. Math. Mothly, 2 (2005) 822 825 7 J. M. Bowei ad G. de Baa, Nested adicals, Ame. Math. Mothly, 98 (99) 735 739