Here are two solutions to the programming exercise to estimate pi^2/6 by
adding the reciprocals of the squares of the integers.

In BASIC:

'parsum.bas
'computes the first ul partial sums
'written 1/1/11 by chip bradley

OPEN "O", #1, "PARSUM.OUT"
INPUT "Upper limit "; ul
FOR p = 1 TO ul
parsum# = parsum# + 1 / p ^ 2
PRINT p, parsum#
PRINT #1, p, parsum#
NEXT p
END

In JavaScript:

<script>
s=0;ul=80000
for (n=1;n<ul;n++) {
 s+=1.0/(n*n)
 document.write('<br>')
 document.write(n + ' ' + s)
 }
</script>

1 1
2 1.25
3 1.3611111111111112
4 1.4236111111111112
5 1.4636111111111112
6 1.4913888888888889
7 1.511797052154195
8 1.527422052154195
9 1.5397677311665408
10 1.5497677311665408
11 1.558032193976458
12 1.5649766384209025
13 1.5708937981842162

...

79993 1.6449215658325393
79994 1.6449215659888128
79995 1.6449215661450822
79996 1.644921566301348
79997 1.6449215664576096
79998 1.6449215666138675
79999 1.6449215667701214 

BTW, going back to the harmonic series (where the denominators are not squared),
the non-convergence is illustrated in the opening pages of John Derbyshire's
"Prime Obsession" as follows.