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\noindent {\bf Problem statement}\footnote*{This series occurs in a
text first published in 1908 by Thomas John l'Anson Bromwich, M.A.,
Sc.D., F.R.S., ``based on courses of lectures given at Queen's
College, Galway''. A knowledge of history is valuable for scholars in
all fields!}~Prove that the series
$$1+ {{a+1}\over {b+1}} + {{(a+1)(2a+1)}\over {(b+1)(2b+1)}} +
{{(a+1)(2a+1)(3a+1)}\over {(b+1)(2b+1)(3b+1)}} + \ldots$$ converges if
$b>a>0$ and diverges if $a\ge b> 0$.










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