Calculus/Proofs

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Expert: - 11/6/2008


Question
Prove that 1+1/4+1/9+...1/n^2 <or= 2-1/n for every positive integer.

Answer
There might be a more elegant way to solve the problem, but this one works.

For n=1, we have 1 = 2 - 1/1, which is 1 = 1, so when n is 1 it is equal.

For n =2, we have 1 + 1/4 <= 2 - 1/2, or 1.25 < 0.5, so it is true at n = 2.

At n=3, we have 1.25 + 1/9 <= 2 - 1/3, or 1.36111... <= 1.666..., so it is true for n = 3.

Here is a list of several proofs that the Σ(1/n²) converges to π²/6.
The are found at secamlocal.ex.ac.uk/people/staff/rjchapma/etc/zeta2.pdf

It is known that π²/6 is approximately 1.644934067.  
2 - π²/6 is 0.355065933, which is greater than 1/3.

I have already stated the first 3 cases and shown that they are true.

For all cases after this, the fraction that is subtracted from 2 is 1/n where n is greater than 3.  We know that all fractions of this form are less than the distance from 2 (which is approximately 0.355065933), so this is true for all n.

An interesting side note:

The answer between 2 and the limit appears to be greater than 1/π and less than 1/e.  I'm not sure that this matters much, but it looks kind of curious.  

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