Prove that the series
converges for all and let
denote the value of this sum for each
. Prove that
is continuous for
and prove that
Proof. First, the series converges for all real by the comparison test since
for all . Therefore, the convergence of
implies the convergence of
. Furthermore, this convergence is uniform by the Weierstrass
-test with
given by
, and again
converges. Thus, by Theorem 11.2 (page 425 of Apostol),
is continuous on the interval . Therefore, we may apply Theorem 11.4 (page 426 of Apostol):
since if
and equals 2 if