Just about to finish my analysis course, but I've having some trouble with a few problems for my final assignment. Any help would be awesome as this course is driving me mad! Thanks so much for the help. You have no clue what a relief resources like this site are!
Forward to the Problem:
In the definition of the integral, we considered partitions of intervals to be arbitrary, but whenever we actually computed one, we used equispaced partitions. The following is a famous example of how to integrate $f(x) = x^t$ for any $t > 0$ directly using non-equispaced partitions. Let $a < b$ be positive numbers, and for any positive integer n, let $r = (b/a)^{1 \over n} > 1$. Then,
$a< ar^0< ar^1< ar^2 < ... < ar^{n-1}< ar^{n}=b$
so if we set $x_i = ar^i$ for $i = 0 ... n$, then $ P_n = \lbrace x_0,...,x_n \rbrace $ is a is a partition of $[a, b]$.
Part A:
Super easy, no need for help here :)
I just had to show that $x^t$ was an increasing function.
Part B:
Using the formula for summing geometric series, show that $L(f, P_n) = a^{t+1}(r-1)(r^{n(t+1)}-1)=(b^{t+1}-a^{t+1})(r-1)/(r^{t+1}-1)$, and $U(f, P_n)=r^t(f,P_n)$
Part C: Use the last problem and part (b) to prove that:
$inf_{n\ge\ 1} U (f,P_n)=sup_{n\ge\ 1}=(b^{t+1}-a^{t+1})/(t+1)$
The last problem:
Let $a_n=\gamma^{s \over n}$ then for $\gamma, s > 0$ and all positive integers $n$:
$\lim_{n\to \infty} a_n = 1$