A geometry problem by Rishabh Jain

Geometry Level 3

$\arctan { \frac { 1 }{ 3 } } +\arctan { \frac { 1 }{ 7 } + } \arctan { \frac { 1 }{ 13 } +\arctan { \frac { 1 }{ 21 } + } } ........upto\quad \infty$ find the above sum .

\frac { \pi }{ 4 }

\frac { \pi }{ 8 }

\frac { \pi }{ 2 }

\frac { \pi }{ 3 }

1 solution

Rishabh Jain
Jun 25, 2014

$the\quad pattern\quad is\quad as\quad follows\quad :\quad 1+1\times 2,1+2\times 3,1+3\times 4,....$ then using the formula of $arctan{ (n+1)-arctan{ (n) } }$ cancelling the terms you will get $arctan{ (n+1)-arctan{ (1) } }$ then apply limit where n approaches $\infty$ you will get the final answer

That's very interesting! The key step is the identity

$ArcTan(\dfrac { 1 }{ { n }^{ 2 }+n+1 } )=$

$ArcTan(n+1)-ArcTan(n)$

I'm pretty sure there's a nice graphical illustration of how this works.

Michael Mendrin - 6 years, 9 months ago

yes sir you are amazingly good at math

samarth sangam - 6 years, 9 months ago

A geometry problem by Rishabh Jain

1 solution

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