Ask Me Help Desk - Upper limit of 1/1+1/2+1/3+.+1/n

upper limit of 1/1+1/2+1/3+.+1/n

how can it be proved that
1/1+1/2+1/3+... +1/n<k*(log of n to the base e),
where k is a sufficiently large positive costant and n is a positive integer.

Take note that $log_{e}(n)=ln(n)$

This appears to be related to the gamma constant.

$\lim_{n\to {\infty}}\left(\sum_{k=1}^{n}\frac{1}{k}-ln(n)\right)$

$=\lim_{n\to {\infty}}\left(\frac{1}{1}+\frac{1}{2}+\frac{1}{3} +....+\frac{1}{n}-ln(n)\right)={\gamma}\approx {.577215664....}$

This can be thought of the difference between the sum and the integral of 1/x.

$=\lim_{n\to {\infty}}\left(\sum_{k=1}^{n}-\int_{1}^{n}\frac{1}{x}dx\right)$

If we let ${\gamma}$ be of secondary importance, we can write it as:

$\sum_{k=1}^{n}\frac{1}{k}-\int_{1}^{n}\frac{1}{x}dx\approx{\gamma}$

${\therefore}, \;\ \sum_{k=1}^{n}\frac{1}{k}\approx \int_{1}^{n}\frac{1}{x}dx+{\gamma}$

If you Google the 'gamma constant', you may find your proof in detail if the hint I gave is not enough.

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Here's another way:

Given that ln(a) is defined as the area under the curve y=1/x from x=1 to x=a, it's straight forward to show that:

ln(N) > 1/2 + 1/3 +1/4 +... +1/N (see figure)

Or:

1+ln(N) > 1+ 1/2 + 1/3 +... +1/N

Since 1+ ln(N) = [1+ln(N)]* ln(N)/ln(N)

If we let k = [1+ln(N)]/ln(N)

You have:

1+ln(N) = k*ln(N) > 1 + 1/2 +1/3 +1... +1/N