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 [L1 ] 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: [L2 ]
Larry Smith I'd just call it one and three quarters and leave it at that. ;)