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A nice but easy proof!

Prove that the value of, $\sum_{n=1}^{n}\dfrac{1}{n^2}<2$.Feel free to post your innovative methods! I have posted mine below!

11 months, 1 week ago

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We basically have to prove that,$\sum_{n=2}^{n}\dfrac{1}{n^2}<1$ $$\text{Motivation of proof}:$$,the first idea that struck me was the use of telescoping series,if i could write,$$\dfrac{1}{n^2}<\text{something}$$ and write,$$\dfrac{1}{(n-1)^2}<\text{another something}$$,and when we add the terms of $$L.H.S$$ we would get the required expression and when we add the terms of the $$R.H.S$$ we would get a telescoping series.The next thing that came to my mind was that,$n^2>(n)(n-1)$(we have taken the greater than sign as then when we take the reciprocal the sign would get reversed),since $$\dfrac{1}{(n)(n-1)}=\dfrac{1}{n-1}-\dfrac{1}{n}$$,and we would get a telescoping series like this, $\dfrac{1}{2^2}<\dfrac{1}{1}-\dfrac{1}{2}\\ \dfrac{1}{3^2}<\dfrac{1}{2}-\dfrac{1}{3}\\ .\\ .\\ .\\ \dfrac{1}{n^2}<\dfrac{1}{n-1}-\dfrac{1}{n}\\ \text{adding,we get}:\\ \sum_{n=2}^{n}\dfrac{1}{n^2}<1-\dfrac{1}{n}<1$.Hence proved.And done! · 11 months, 1 week ago

I like your motivations haha :P · 11 months, 1 week ago

From where you got this motivation ? · 11 months, 1 week ago

I am sorry but i don't understand the meaning of your comment.Could you please explain? · 11 months, 1 week ago

Simplification: Source of the question = ? · 11 months, 1 week ago

Arihant Mathematical Olympiads. · 11 months, 1 week ago

Well, yours is the simplest way. For the sake of mentioning, one can also write a proof by induction for this. · 11 months, 1 week ago

How do you do a proof by induction (that is fundamentally different from his approach)? Staff · 11 months, 1 week ago

@Calvin Lin Oops, although Induction is tempting at first look, it isn't the best way to go about. I haven't found an inductive proof yet 😕. · 11 months, 1 week ago

Right. The inductive proof that I know, is to show that for $$n \geq 2$$,

$\sum_{i=1}^n \frac{1}{i^2 } < 2 - \frac{1}{n}.$

This is similar to what Adarsh did. Staff · 11 months, 1 week ago

from the basel problem we get $\sum_{n=1}^\infty (\frac{1}{n^2}) = \frac{\pi^2}{6}<2$. although this is probably not intended, but still works as a good proof. we prove the basel proble: Euler's original derivation of the value $$\frac{π^2}{6}$$ essentially extended observations about finite polynomials and assumed that these same properties hold true for infinite series.

recall the Taylor series expansion of the sine function

$\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots.$ Dividing through by x, we have

$\frac{\sin(x)}{x} = 1 - \frac{x^2}{3!} + \frac{x^4}{5!} - \frac{x^6}{7!} + \cdots.$ Using the Weierstrass factorization theorem, it can also be shown that the left-hand side is the product of linear factors given by its roots, just as we do for finite polynomials

\begin{align} \frac{\sin(x)}{x} &= \left(1 - \frac{x}{\pi}\right)\left(1 + \frac{x}{\pi}\right)\left(1 - \frac{x}{2\pi}\right)\left(1 + \frac{x}{2\pi}\right)\left(1 - \frac{x}{3\pi}\right)\left(1 + \frac{x}{3\pi}\right) \cdots \\ &= \left(1 - \frac{x^2}{\pi^2}\right)\left(1 - \frac{x^2}{4\pi^2}\right)\left(1 - \frac{x^2}{9\pi^2}\right) \cdots. \end{align} If we formally multiply out this product and collect all the x2 terms (we are allowed to do so because of Newton's identities), we see that the x2 coefficient of sin(x)/x is

$-\left(\frac{1}{\pi^2} + \frac{1}{4\pi^2} + \frac{1}{9\pi^2} + \cdots \right) = -\frac{1}{\pi^2}\sum_{n=1}^{\infty}\frac{1}{n^2}.$ But from the original infinite series expansion of sin(x)/x, the coefficient of x2 is −1/(3!) = −1/6. These two coefficients must be equal; thus,

$-\frac{1}{6} = -\frac{1}{\pi^2}\sum_{n=1}^{\infty}\frac{1}{n^2}.$ Multiplying through both sides of this equation by $$-\pi^2$$ gives the sum of the reciprocals of the positive square integers.

$\sum_{n=1}^{\infty}\frac{1}{n^2} = \frac{\pi^2}{6}.$ · 11 months, 1 week ago

Thanks a lot for adding the proof! · 11 months, 1 week ago

You are right,it wasn't intended but if you write this,you are expected to prove it too. · 11 months, 1 week ago

fine · 11 months, 1 week ago

Isn't this reimann zeta(2)? Then the value will be 1.64 only. · 10 months ago

Yes,it is but that isn't the intended proof,or else you will have to prove how you found the value of $$\zeta{(2)}$$. · 10 months ago