i want to calculate the integral of sine(x) using riemann sums and got the stuck with the series.
the process scan is attached and shows the problem clearly.
Make it more generally:
Let's compute Let's use the following partition thus, our limit becomes Now, use the formula and the limit is
now let's turn this sum into a telescoping one,
equals and equals (Note that by adding you'll get the sum of the limit.)
Finally, by taking the limit, we have
and
as thus the limit equals and we're done.
(You just need to put and there's your sum.)