A sequence {A(n)} satisfies:

A(1) = √2, and:

A(n)= (√2)^A(n-1) for n =2, 3,……

Determine this limit:

Limit A(n)
n → ∞

Iterating we see it approaching 2. Also, as an iterative way to solve a=(sqrt(2))^a, it makes sense that it does approach 2, as that is the solution to this equation.

2       1.63252691943815284477349538102471960207910885703
3       1.76083955588002809075664989563837274807980409428
4       1.840910869291010298414834540976459261499504581703
5       1.892712696828510480823454616483161041843956500151
6       1.926999701847100431015638317518107700700545351688
7       1.950034773805817581828743269620326268525916688118
8       1.965664886517318713890147612742296171566475769559
9       1.97634175440970237769634062487778508178583046788
10      1.983668399303821147193751746120294572513188717376
11      1.988711773413953318328100793390310129015115887576
12      1.992190882947057099754305148437942178137452080847
13      1.994594450712101177628954134628009966698438518787
14      1.996256666265858383796961721984270499249655649735
15      1.997407001141335832237269444963677747430091789283
16      1.998203477508701623889882900838052689588142118409
17      1.99875513308459188513700733450289408807345229695
18      1.999137310119390933251325219858478209309442937077
19      1.999402118324996572890407660901023010928440698227
20      1.999585622935681030440026876488145443030969515032
21      1.9997127963296400677615979624895295319818257781
22      1.999800935492970439267454158427908364363541353084
23      1.999862023757782662903984865573590037514896011239
24      1.99990436444333561544223259662939032715437171446
25      1.999933711582099661953497557433590047243516111054
26      1.999954052897820734463148555833396073320003396208
27      1.999968152149243633757769656661098363594733201571
28      1.999977924873870025693166508189821601314066440079
29      1.999984698747094886788765666158756849635291251376
30      1.999989394007811653100223109499661581401137302835
31      1.999992648499928777975410743365691329839457823063
32      1.999994904334944207832832399896935817683812056215
33      1.999996467957252334987702171758189039459745527488
34      1.999997551776026290173582083482586477560793906736
35      1.999998303021175178458360828799070869174866078896
36      1.999998823744257999547343823890033132343833241089
37      1.999999184681815000920659606985350591662976062226
38      1.999999434864578653162415444760035786919253282786
39      1.999999608278014420416303600498716542678399347884
40      1.999999728479028563079157484431904431238451953404
41      1.999999811796013040786608125627954751216868195202
42      1.999999869546941323580837205119954870619759765704
43      1.999999909576832227106325637947647817912045683786
44      1.999999937323437183007742675084725061154375095207
45      1.999999956555917668060592642701560561966764622721
46      1.999999969886857046302412648721800431606909361955

 5   point 10
10   A=sqrt(2)
20   for Iter=2 to 46
30    A=sqrt(2)^A:print Iter,A
40   next

Posted by Charlie on 2011-09-24 14:57:05