<s>
In	O
mathematical	O
analysis	O
,	O
Tannery	B-Algorithm
's	I-Algorithm
theorem	I-Algorithm
gives	O
sufficient	O
conditions	O
for	O
the	O
interchanging	B-Algorithm
of	I-Algorithm
the	I-Algorithm
limit	I-Algorithm
and	I-Algorithm
infinite	I-Algorithm
summation	I-Algorithm
operations	I-Algorithm
.	O
</s>
<s>
Tannery	B-Algorithm
's	I-Algorithm
theorem	I-Algorithm
follows	O
directly	O
from	O
Lebesgue	O
's	O
dominated	O
convergence	O
theorem	O
applied	O
to	O
the	O
sequence	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Tannery	B-Algorithm
's	I-Algorithm
theorem	I-Algorithm
can	O
be	O
used	O
to	O
prove	O
that	O
the	O
binomial	O
limit	O
and	O
the	O
infinite	O
series	O
characterizations	O
of	O
the	O
exponential	O
are	O
equivalent	O
.	O
</s>
