Reducing Leibniz perl code in characters

Perl 53 bytes

You can save two more strokes:

<>;print${\map$a-=(-1)**$_/(2*$_-1),1..$_},$a=$/for<>

<>;map($s-=(-1)**$_/(2*$_-1),1..$_),$s=!print$s,$/for<>

Without using any Perl non-standard features (such as say). 3 bytes are wasted removing the first value from the input.

Sample usage:

in.dat

2
10
20

$ perl leibniz.pl < in.dat
0.760459904732351
0.77290595166696

map{$s-=(-1)**$_/($_*2-1)for 1..<>;$s=!print$s,$/}1..<>

The alternative may be run interactively, without requiring an input file. Which, as I understand, is what you meant by running indefinitely.