Nested Radical Constant

Arjen Markus (7 november 2006) Reading "Mathematics by Experiment" by Borwein and Bailey and its companion volume "Experimentation in Mathematics", I was reminded of a series of numbers I was fascinated by a long time ago:

    x1 = 1
    x2 = sqrt(1+sqrt(2))
    x3 = sqrt(1+sqrt(2+sqrt(3)))
    ...

It seemed to converge very rapidly, but at the time I only had a handheld calculator at my disposal. I forgot about for years and now it drifted up again. So I wrote a little script to examine the series.

I found that with the usual double precision arithmetic only 20 or 21 iterations were enough to exhaust the precision. So I rewrote it to use math::bigfloat instead. Here is the script:

 # sqrtseries.tcl --
 #     Generate a series based on the following
 #     formula:
 #     x1 = 1
 #     x2 = sqrt(1+sqrt(2))
 #     x3 = sqrt(1+sqrt(2+sqrt(3)))
 #     ...
 #
 #     Note: just curious to see if it converges or not and to what
 #
 package require math::bigfloat
 namespace import ::math::bigfloat::*

 proc sqrtSeries {n} {
     set r [expr {$n}]
     while { $n > 1 } {
         set r [expr {$n-1+sqrt($r)}]
         incr n -1
     }
     expr {sqrt($r)}
 }

 proc sqrtSeriesB {n} {
     set r [fromstr $n.0]
     while { $n > 1 } {
         set r [add [fromstr [expr {$n-1}]] [sqrt $r]]
         incr n -1
     }
     sqrt $r
 }

 set tcl_precision 17
 puts "Explicitly:"
 puts "1: [expr {sqrt(1)}]"
 puts "2: [expr {sqrt(1+sqrt(2))}]"
 puts "3: [expr {sqrt(1+sqrt(2+sqrt(3)))}]"
 puts "4: [expr {sqrt(1+sqrt(2+sqrt(3+sqrt(4))))}]"

 for { set n 1 } { $n < 150 } { incr n } {
    puts "$n: [tostr [sqrtSeriesB $n]]"
 }

The resulting output is:

 1: 1.0
 2: 1.5537739740300374
 3: 1.7122650649295326
 4: 1.7487627132551438
 1: 1.0
 2: 2.
 3: 1.71
 4: 1.749
 5: 1.7562
 6: 1.7576
 7: 1.75788
 8: 1.75793
 9: 1.757932
 10: 1.7579326
 11: 1.7579327
 12: 1.75793275
 13: 1.75793276
 14: 1.7579327566
 15: 1.75793275661
 16: 1.75793275662
 17: 1.757932756618
 18: 1.75793275661800
 19: 1.75793275661800
 20: 1.757932756618004
 21: 1.7579327566180045
 22: 1.75793275661800453
 23: 1.757932756618004533
 24: 1.7579327566180045327
 25: 1.75793275661800453271
 26: 1.757932756618004532709
 27: 1.7579327566180045327088
 28: 1.75793275661800453270882
 29: 1.75793275661800453270882
 30: 1.757932756618004532708820
 31: 1.7579327566180045327088196
 32: 1.75793275661800453270881964
 33: 1.757932756618004532708819638
 34: 1.7579327566180045327088196382
 35: 1.75793275661800453270881963822
 36: 1.757932756618004532708819638218
 37: 1.7579327566180045327088196382181
 38: 1.75793275661800453270881963821814

 ...

 136: 1.7579327566180045327088196382181385276531999221468377043101355003851102326
 74446757572344554000259452970932471847826956725286405867741108546115435
 137: 1.7579327566180045327088196382181385276531999221468377043101355003851102326
 744467575723445540002594529709324718478269567252864058677411085461154351
 138: 1.7579327566180045327088196382181385276531999221468377043101355003851102326
 74446757572344554000259452970932471847826956725286405867741108546115435117
 139: 1.7579327566180045327088196382181385276531999221468377043101355003851102326
 744467575723445540002594529709324718478269567252864058677411085461154351167
 140: 1.7579327566180045327088196382181385276531999221468377043101355003851102326
 7444675757234455400025945297093247184782695672528640586774110854611543511675
 141: 1.7579327566180045327088196382181385276531999221468377043101355003851102326
 744467575723445540002594529709324718478269567252864058677411085461154351167460
 142: 1.7579327566180045327088196382181385276531999221468377043101355003851102326
 7444675757234455400025945297093247184782695672528640586774110854611543511674597
 143: 1.7579327566180045327088196382181385276531999221468377043101355003851102326
 74446757572344554000259452970932471847826956725286405867741108546115435116745975 
 144: 1.7579327566180045327088196382181385276531999221468377043101355003851102326
 74446757572344554000259452970932471847826956725286405867741108546115435116745974
 8
 145: 1.7579327566180045327088196382181385276531999221468377043101355003851102326
 74446757572344554000259452970932471847826956725286405867741108546115435116745974
 83
 146: 1.7579327566180045327088196382181385276531999221468377043101355003851102326
 74446757572344554000259452970932471847826956725286405867741108546115435116745974
 8276
 147: 1.7579327566180045327088196382181385276531999221468377043101355003851102326
 74446757572344554000259452970932471847826956725286405867741108546115435116745974
 82765
 148: 1.7579327566180045327088196382181385276531999221468377043101355003851102326
 74446757572344554000259452970932471847826956725286405867741108546115435116745974
 827650
 149: 1.7579327566180045327088196382181385276531999221468377043101355003851102326
 74446757572344554000259452970932471847826956725286405867741108546115435116745974
 8276498

I ran this sequence of digits through Sloane's Online Encyclopedia of Integer Sequences [1 ] and that site came up with the name of this particular constant: "Nested radical constant".

After all these years (20 or more :)) I finally have a name for this thing.

More information at: [2 ]

Larry Smith I'd just call it one and three quarters and leave it at that. ;)