Without allowing functions to take on infinite values, such essential results as the monotone convergence theorem and the dominated convergence theorem would not make sense.
12.
:: : : OK, that argument works for monotone sequences, but won't work here for the same reason the monotone convergence theorem won't work here.
13.
The Lebesgue integral is better able to describe how and when it is possible to take limits under the integral sign ( via the powerful monotone convergence theorem and dominated convergence theorem ).
14.
Indeed, since each term of the binomial expansion is an increasing function of " n ", it follows from the monotone convergence theorem for series that the sum of this infinite series is equal to " e ".
