The techniques of calculus enable us to sum a lot of series involving binomial coefficients. This is the subject of this section.

Suppose that we have to evaluate the sum given by

From now on, to avoid clutter, we’ll write as simply , where the upper index should be understood to be present. Thus,

This series can be generated using a manipulation involving differentiation, as follows:

Consider the binomial expansion

If we differentiate both sides with respect to , look at what we’ll obtain:

Now, all that remains is to substitute , upon which we obtain:

This is what we were looking for. Thus,

Had we substituted , we would’ve obtained

Thus, we have evaluated another interesting sum.

Suppose that we now wish to evaluate given by

The alert reader would immediately realize that integration needs to be applied here. How exactly to do so is now described. Consider again the expansion.

If we integrate this with respect to , between some limits say to , we obtain

To generate the sum a little thought will show that we need to use , so that we obtain

Thus, equals

Try some other values for and and hence generate other series on your own. Be as varied as you can in choosing these limits.