The binomial coefficient is defined by
where and
.
- For
show that
- Consider the series whose terms are defined by
Prove that
- We compute each of the requested quantities using the given definition, where
,
- Proof. The proof is by induction. For
we have
For
we have
Thus, for
we have
Hence, the statement holds for the case
. Assume then that the statement holds for
. Then,
Furthermore, from the definition of
we have
But,
for all positive integers
, and
by the induction hypothesis. Thus,
. This gives us the first part of the statement,
for all positive integers
. Next, since
we have
Hence, both statements are true for all positive integers
“Furthermore, from the definition of a_n we have” the k in the fraction should be k+1