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Hello , everyone , I need help . Does the following have definite value or not , if yes , then what is it ?

\[1+\frac{1}{4} +\frac {1}{9} +\frac {1}{16}+\frac{1}{25} . . . \infty\]

Reshare to know the answer.

Note by Utkarsh Dwivedi
2 years, 11 months ago

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

Siddhartha Srivastava - 2 years, 11 months ago

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Thanks , Bro. Well, just asking that have you realised it's implications that it has a fixed value ?

Utkarsh Dwivedi - 2 years, 11 months ago

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I don't understand what you mean. I don't see any very obvious implication of this fact.

Siddhartha Srivastava - 2 years, 11 months ago

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@Siddhartha Srivastava Not going in too different direction , but , once , when this thing came to my mind then , one of my colleagues got up with an idea.

Here's the idea : Suppose there is a line of massive objects ( perhaps black holes or stars or whatever you may want ) , such that each object touches the adjacent two objects . All objects in the line have a fixed mass, let's say \(M\). And they all are arranged in a straight line. They all are spherical with a fixed diameter \(d\) and are touching each other. There are infinite objects arranged in a endless line. Remember , they are in absolute space without any influence of any force . What we have to do is with their gravitational force . A simple question : What is the total force acting on each of the objects ( in their center of mass , perhaps ) by the gravity of all the infinite objects on one of its side . My approach :

By Law of Gravitation , total force exerted by all the objects : \[\frac {GMM_{1}}{r^{2}} + \frac {GMM_{2}}{(2r)^{2}} + . . . + \frac {GMM_{\infty}}{(nr)^{2}}\] Where mass of our selected object is \(M\) . And r is the distance. Now as mass of all object is the same and the distance is or should be a multiple of \(d\) , clearly , it is equal to : \[\frac {GM^{2}}{d^{2}}(1 +\frac {1}{2^{2}}+\frac {1}{3^{2}} . . .) \] And so as we have both discovered the sum of the infinite sequence is finite or has a fixed value , so we can say the force of gravity would be finite , even though it seems to be infinite . If you haven't understood just try to visualize .

Utkarsh Dwivedi - 2 years, 11 months ago

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Pi^2/6. For a solution you can go to plus.mah.org which is a very good website explaining MATHS and PHYSICS.

Deepak Pant - 2 years, 10 months ago

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Yeah, I've had that , well, the thing is that , you see, my comment to @Siddhartha Srivastava about the gravity and all , is that true ? , this note was created to understand that concept. Any ideas ?

Utkarsh Dwivedi - 2 years, 10 months ago

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@Michael Mendrin , Is my approach correct ?

Utkarsh Dwivedi - 2 years, 11 months ago

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@Calvin Lin

Utkarsh Dwivedi - 2 years, 11 months ago

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