Example: 13  
Find the sum of the series
if you are told that this corresponds to an expansion of a binomial, of the form .

Solution: 13  
We need to determine and . For that, we can compare the terms of this series with the corresponding terms in the following general expansion.
Thus,
Solving for and from these two equations, we get and . Thus, the sum of the series is

Example: 14  
Find the sum of the series
for

Solution: 14  
Before solving this problem, ponder a moment over the following fact:
In the expansion of , if , that is, if is much smaller than , then the expansion can be approximated as
since all higher order terms can be neglected due to the small magnitude of.
Coming to the problem, note that if , i.e, if , then,
Thus, the sum of the series is (to a good approximation)

Example: 15  
Evaluate correct to four decimal places. 
Solution: 15  
We have
Note that we only considered the first three terms of the expansion because the higher order terms would not have had any effect on the answer up to the fourth decimal place.
