This problem is a reprise of problem 6 with 1.001 replaced by 1.000001

Compute an approximation to $$\int_1^{3}\frac{1}{x}\,dx\,,$$ which gives the area under $y=\frac{1}{x}$ for $1\le x\le 3$, using a modified Riemann sum with the (NOT equally spaced) partition $$1,1.000001,1.000001^2,1.000001^3,\dots,1.000001^N,3$$ and left hand endpoints EXCEPT neglecting the area of the last rectangle. Here $N$ denotes the largest possible power which fits in the interval $1\le x\le 3$.

Please note that the problem is NOT asking for the value of $\int_1^{3}\frac{1}{x}\,dx$. Rather it is asking for the EXACT values of the areas of approximating rectangles and for the EXACT value of the sum of the areas of the rectangles. Calculator approximations (no matter how accurate) will NOT be accepted. Do the calculations by hand using fractions (until you notice the pattern in the areas).

The number of approximating rectangles is:
$N$ =
The area of the first rectangle =
The area of the second rectangle =
The area of the third rectangle =
The sum of the areas of the $N$ rectangles =

The ingenious idea of using these unusual nonequally spaced partitions to compute the area under $y=\frac{1}{x}$ thus relating it to Napier's logarithms is due to a Belgian monk, Gregory St. Vincent around 1647.