During tutorials last week, a number of students asked how to understand identities that are used in the calculation of various Riemann sums and their limits.
These identities are:
Let’s go through these one by one. We must first remember what the sigma notation means. If we have:
It means the sum of terms of the forms f(i) for i starting with 1 and going up to i=n. Sometimes n will actually be an integer, and sometimes it will be left arbitrary. So, the above sum can be written as:
We haven’t specified what f is, but that’s because this statement is general and applies for any time of function of i. In the first of the identities above, the function is simply f(i)=1, which isn’t a very interesting function, but it still is one. It says, whatever i we put in, output 1. So this sum can be written as:
Where there are n terms.…