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.

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

 Category Mathematics