Applying Abel's lemma on summation by parts,
we shall investigate partial sums of the
bilateral hypergeometric series ($\Re(c+d-a-b)>1$)
\[{_2H_2}\ffnk{cc}{1}{a,\+b}{c,\+d}
\:=\:\sum_{k=-\infty}^{+\infty}\frac{(a)_k(b)_k}{(c)_k(d)_k}
\:=\:\Gam\fnk{c}{1-a,1-b,c,d,c+d-a-b-1}{c-a,\:d-a,\:c-b,\:d-b}\]
due to Dougall (1907). Several transformations will be established
which will then be utilized to show the existence of numerous infinite
series expressions for $\pi$. Three typical ones can be reproduced as follows
\bnm
\frac{4}{\pi}
&=&\sum_{k=0}^{\infty}
\fnk{ccc}
{\frac12,\+\frac12,\+\frac12}{\rule[1mm]{0mm}{3mm}1,\+1,\+1}_k
\frac{1+6k}{4^k},\\
\frac{8}{\pi}
&=&\sum_{k=0}^{\infty}(-1)^k
\fnk{ccc}
{\frac12,\+\frac14,\+\frac34}
{\rule[1mm]{0mm}{3mm}1,\+1,\+1}_k
\frac{3+20k}{4^k},\\
\frac{16}{\pi}\!
&=&\sum_{k=0}^{\infty}
\fnk{ccc}
{\frac12,\+\frac12,\+\frac12}{\rule[1mm]{0mm}{3mm}1,\+1,\+1}_k
\frac{5+42k}{64^k}.
\enm
They are originally discovered by Ramanujan (1914) with each
of them representing a large class of similar formulae for $\pi$.