[hxtefuon] alternating series for pi from its continued fraction convergents

we start with the simple continued fraction of pi, then compute its convergents.  subtracting consecutive convergents yields an alternating series.

unlike other famous alternating series for pi (e.g., Ramanujan, Chudnovsky, arctangent), you (probably) can't calculate these terms without calculating pi first by some other means beforehand.

by Lochs's Theorem, each term improves precision by about one base-10 digit.  this is not very good compared to other alternating series.  this is a little surprising, as we started with continued fraction convergents which are optimal in the sense of being the best fraction for a given size of denominator.

the numerators are all 1 (Egyptian fraction) probably because continued fraction is a greedy algorithm.

? c=contfrac(Pi); n=matsize(c)[2]; u=contfracpnqn(c,n); for(i=2,n, j=i-1; print(u[1,i]/u[2,i]-u[1,j]/u[2,j]))

3 + 1/7 - 1/742 + 1/11978 - 1/3740526 + 1/1099482930 - 1/2202719155 + 1/6600663644 - 1/26413901692 + 1/96840976853 - 1/496325469560 + 1/2346251883960 - 1/44006595799206 + 1/1345586183756654 - 1/4127747481719463 + 1/10251870941174304 - 1/44575430382887456 + 1/276132882598044178 - 1/1593289693963483866 + 1/9302537452424752764 - 1/31752770150883945868 + 1/3804187014684187924672 - 1/648286971332373180561600 + 1/1952514197957970456867225 - 1/4875556759407158590614514 + 1/126860029519268053447424974 - 1/6063730539744000159560443801 + 1/247602482617403339909076561758 - 1/3547242210497060338170061973634 + 1/18800940598978810231948005689958 - 1/204185682530545088789414517822446 + 1/2902905707667554501886431715545780 - 1/115314349980593838601889295653199380 + 1/70380893109222621997560772136450259431 - 1/7049524166811053144158347587725348957320 + 1/21290753793293205525523880648516438240520 - 1/148610152181113901248574367224661384427658 + 1/2230144712753445639681849940885168524295394 - 1/45238711379157248420014063216726324300591037 + 1/759006164891627935764147156892282472623777282 - 1/4794566735134355124278397328529273509171050758 + 1/10490893125862832321789768432685940720630516047 - 1/126417015713302320129284264029799172318723557733 + 1/6743567251119057455360928791062337935899394821485 - 1/61722762406209175573917215647732287006910690695510 + 1/546777076463486842522191915680448133024714376670258 - 1/4861255143706449087313512121834381022317101144104659 + 1/15815906530919828851827284396873386717577247524504421 - 1/152822310780406383221046024127581791824820499838462104 + 1/3732566154173503665927183706103777061148583706792158296 - 1/30975954210267369528087864730966858500331494237311153657 + 1/101415778894404286568319715228529899700241296756075976475 - 1/393442633159841464623295556172386497540569789273369232100 + 1/923520957556155674335026752822084415381335776562196852084 - 1/20231406297333367478494604552082014544528144476796508929792 + ...