Prove this:
using Riemann sums
(Technical note: I'm assuming that you are assuming that you know that the integral exists, and you just want to calculate it's value using Riemann Sums, i.e we don't care about proving that it is Riemann integrable directly from the definition... it's probably doable but more work obviously)
So let
be the rectangular left hand approximation with n evenly spaced intervals.
)
)
)
Now the sum should be easy to evaluate.
Here is an extension exercise you may want to try:
a) Let
be a natural number,
=\sum_{k=1}^n k^m)
. It is well known that
)
is a polynomial in
of degree
with leading coefficient (i.e coefficient of
) equal to \frac{1}{m+1}. Use this to evaluate
for natural number m using Riemann sums. (a=0,b=1 is a good enough exercise)
b) Prove the claim given in part a (hint,
-S(n)=(n+1)^m)
, now expand and equate coefficients)
Also, what's the difference between
and
The former is the definition of the latter (infinite sums don't really make sense in the reals, but limits do)
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