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