Let $b>1$ be a real number and let $f$ be given on the interval $[1,b]$ by $f(x)=x^2$. A partition $P_n$ are defined for every postive whole number $n$ as follows:
$P_n: x_0=1, x_1=b^{\frac{1}{n}}, x_2=b^{\frac{2}{n}}+...+x_{n-1}=b^{\frac{n-1}{n}}, x_n=b^{\frac{n}{n}}=b $.
Show that the upper- and lower Riemann sum is given by:
$U(f,P_n)=b^{\frac{2}{n}}(\frac{b^{1/n}-1}{b^{3/n}-1})(b^3-1)$ and $L(f,P_n)=(\frac{b^{1/n}-1}{b^{3/n}-1})(b^3-1)$
Hint: Use the formula $\sum_{i=1}^{n}b^{3(i-1)/n}=\frac{b^{3}-1}{b^{3/n}-1}$
What I have tried so far:
$U(f,P_n)=f(u_1)\Delta x_1+f(u_2)\Delta x_2+f(u_3)\Delta x_3+...+$ $=1^2(b^{1/n}-1)+b^{2/n}(b^{2/n}-b^{1/n})+b^{4/n}(b^{3/n}-b^{2/n})+b^{6/n}(b^{4/n}-b^{3/n})+...+b^{\frac{2n-2}{n}}(b-b^{\frac{n-1}{n}})$
$=b^{1/n}-1+b^{4/n}-b^{3/n}+b^{7/n}-b^{6/n}+b^{10/n}-b^{9/n}+...+b^{3n-2/n}-b^{3n-3/n}$
From here, I am not able to see any "obvious" traits on the sequence