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PMI (not neccesarily) Problem

To prove that \[\frac{1}{3^2} + \frac{1}{5^2} + \frac{1}{7^2} + \cdots +\frac{1}{(2n+1)^2} < \frac{1}{4} \] for all \(n\ belongs\ to\ N\)

This was given to be proved by induction in "Self-Learning Exercises - GSEB".

Note by Megh Parikh
3 years, 4 months ago

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Overkill: Here \(\sum\) indicates \(\sum_{i=1}^\infty\). Since \(\sum \frac{1}{i^2} = \frac{\pi^2}{6}\), we have \(\sum \frac{1}{(2i)^2} = \frac{\pi^2}{24}\). Subtracting the two, we have \(\sum \frac{1}{(2i-1)^2} = \frac{\pi^2}{8}\), thus \(\sum \frac{1}{(2i+1)^2} = \sum \frac{1}{(2i-1)^2} - \frac{1}{1^2} = \frac{\pi^2}{8} - 1 < \frac{1}{4}\).

I'm pretty sure the method is to figure out some always positive \(f(n)\) such that \(f(n) - \frac{1}{(2n+1)^2} > f(n+1)\), so we can induct it on \(\sum_{i=1}^n \frac{1}{(2i+1)^2} < \frac{1}{4} - f(n)\). Ivan Koswara · 3 years, 4 months ago

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@Ivan Koswara That is indeed an overkill. The problem is direct if you know that fact, and hence lets look for solutions which do not directly involve it.

Any suggestions for what \(f(n) \) could be? Calvin Lin Staff · 3 years, 4 months ago

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@Ivan Koswara By guesswork I obtained one f to be \[f(n)=\dfrac{n+\frac{19}{16}}{(2n+1)^2}\] Megh Parikh · 3 years, 4 months ago

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@Megh Parikh Explaining my guesswork:

I assumed f to be rational function with denominator to be (2n+1)^2 and numerator to be linear ax+b

Then I got quadratic in n and made it into linear and got a=1.

Then solving for b in term of n, I got b>9/8

Also \(f(1) - f(n+1) < 1/4 \implies b<5/4\)

So I took b=9.5/8 Megh Parikh · 3 years, 4 months ago

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@Megh Parikh Nicely done.

Note that the second condition is superfluous, because we can choose a different starting point. For example, if we take \( b = 2 \), then with \( g(n) = \frac{n+2}{(2n+1)^2} \), we get that \[ g(n) - g(n+1) > \frac{ 1}{ (2n+1) } ^2 \text{ for } n \geq 1 .\]

It remains to find an \(n\) which would give us

\[ \frac{1}{3^2 } + \frac{1}{5^2} + \ldots + \frac{1}{(2n+1)^2} < \frac{1}{4} - g(n) .\]

In this particular case, \( n = 5 \) works (using a calculator). Calvin Lin Staff · 3 years, 4 months ago

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@Ivan Koswara Thanks Megh Parikh · 3 years, 4 months ago

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@Ivan Koswara I didn't understand the subtraction of the summation. How did you get the denominator \({(2i-1)}^2\) ? .Could you explain better this part? Carlos E. C. do Nascimento · 3 years, 4 months ago

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@Carlos E. C. do Nascimento \(\sum \frac{1}{i^2} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \ldots\), and \(\sum \frac{1}{(2i)^2} = \frac{1}{2^2} + \frac{1}{4^2} + \frac{1}{6^2} + \ldots\), so subtracting the two gives \(\frac{1}{1^2} + \frac{1}{3^2} + \frac{1}{5^2} + \ldots = \sum \frac{1}{(2i-1)^2}\). Ivan Koswara · 3 years, 4 months ago

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\[\frac11^2+\frac13^2+\frac15^2+\dots=\left(\frac11^2+\frac12^2+\frac13^2+\dots\right)-\left(\frac12^2+\frac14^2+\frac16^2+\dots\right)=\frac{\pi^2}{6}-\frac{\pi^2}{4\cdot6}=\frac{\pi^2}{8}\]

Hence,

\[\frac13^2+\frac15^2+\dots=\frac{\pi^2}{8}-1<\frac14\]

Since \(f(n)\) is increasing monotonically and the inequality holds asymptotically, the inequality holds for each \(n\). Cody Johnson · 3 years, 4 months ago

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@Cody Johnson That is exactly my solution above, which is also an overkill (you have to prove \(\sum \frac{1}{i^2} = \frac{\pi^2}{6}\), for example). Ivan Koswara · 3 years, 4 months ago

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