Prove U can't get an constant.

Let us make a series. $S=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}..............$ Prove that this will not converge to a constant.

Extra Credit: Do it without use of calculus.

#NumberTheory

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
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`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
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Comments

This is popularly known as the divergence of the Harmonic Series

Agnishom Chattopadhyay - 6 years, 8 months ago

The proof that I like is by contradiction.

Suppose $S$ converges to a finite number. Then, show that $S = S + 1$ .

(details to follow).

Calvin Lin Staff - 6 years, 6 months ago

It can be done using RATIO TEST.

Sandeep Bhardwaj - 6 years, 8 months ago

Please give the details or any link from where I can learn it.. Thanks.

Satvik Golechha - 6 years, 8 months ago

http://en.wikipedia.org/wiki/Ratio_test

Sandeep Bhardwaj - 6 years, 8 months ago

@Sandeep Bhardwaj – But the limit L here is 1. Therefore, test becomes inconclusive

A Former Brilliant Member - 6 years, 6 months ago

Let us make a AM HM inequality. 1/(a-1) + 1/a + 1/(a+1) >= 3/a So we will take its minnimum value... as 3/a Now lets put the values . Leaving 1 then taking 1/2 , 1/3, 1/4 then next three....then next three... So we will get its min value as 1 + 1 + 1/2 + 1/3. ..... to infinity. Then again doing it. 1+1+1+1+1+1+..................to infinite +1/2 + 1/3 + 1/4 .... So its minnimum value comes out to be infinite .

Soorry can't make the solution latex ...!

Utsav Singhal - 6 years, 8 months ago

Even though the harmonic series does not converge, Is it possible for a partial sum of the series to be an integer?

Brian Charlesworth - 6 years, 6 months ago

2.30 approx

Kanhaiya Yadav - 6 years, 8 months ago

No no, it is totally wrong . There is no value of it . It will be infinite

Utsav Singhal - 6 years, 8 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$