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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!}~Suppose $\displaystyle \alpha_n = {1\over {n^2}}$ except
when $n$ is a square, and $\displaystyle \alpha_n = {1\over
{n^{3/2}}}$ when $n$ is a square.  The series whose $n^{\rm th}$ term
is $\alpha_n$ is therefore:
$$1 + {1\over {2^2}} + {1\over {3^2}} + {1 \over {4^{3/2}}} + {1\over
{5^2}} + {1\over {6^2}} + {1\over {7^2}} + {1\over {8^2}}+ {1 \over
{9^{3/2}}}+ \ldots $$ Does this series converge?

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