Saturday, 9 August 2014
CHAPTER 6 - Binomial Involving Calculus
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
From now on, to avoid clutter, we’ll write
, 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
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
, we obtain
To generate the sum
a little thought will show that we need to use
, so that we obtain
Try some other values for
and hence generate other series on your own. Be as varied as you can in choosing these limits.
August 09, 2014
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