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OMG! can someone plz latexify it for me? Thanks

Weierstress proof for the second Smile

Let me prove a genl inequality first whose's case is this sum:
To prove $rac {1 cdot 3 cdot 5........ cdot (2n-1)}{2 cdot 4 cdot 6...........cdot 2n} > rac {1}{n+1}$

let $a_{1} = rac {1}{2}, a_{2} = rac {1}{4}....., a_{n} = rac {1}{2n} . s_{n} = sum_{i=1}^{n} a_{1} < 1$

Applying Weierstress's inequality,
$(1+s_{n}) < (1+a_{1}) cdot (1+a_{2}) cdot (1+a_{3}).... cdot (1+a_{n}) implies (1+ rac {1}{2}) cdot (1+ rac {1}{4}) cdot (1+ rac {1}{6}) cdot........ (1+ rac {1}{2n}) cdot > 1 + rac {1}{2} + rac {1}{4} + rac {1}{6}........... + rac {1}{2n} implies LHS > 1 + 2^{-1} + 4^{-1} ........ + (2n)^{-1}. implies rac {3}{2} cdot rac {5}{4} cdot ........ rac {2n+1}{2n} cdot > 1 + 2^{-1} + 4^{-1} ........ + (2n)^{-1}.$ --->1

Arithmetic mean s of -1th powers > -1th power of AM

But $rac {2^{-1} + 4^{-1} ........ + (2n)^{-1}}{n} > ( rac {( 2+4+6+...+2n)}{n})^{-1}$

On simplification you get

$1 + 2^{-1} + 4^{-1} ........ + (2n)^{-1} > rac {2n+1}{n+1}$ ---> 2

from 1 and 2

$implies rac {3}{2} cdot rac {5}{4} cdot ........ rac {2n+1}{2n} cdot > rac {2n+1}{n+1}$

Hence on simplification we get the result Smile

So after that apply n = 1998 Smile

hsbhatt will post the soln later ok? Am busy now :D

i dont know how to use latex properly Help?

also 2nd part can be done by Wierstrass

ask all your doubts to pardesi da, he gave me this sum :D

no i dont think so

computer001 i appreciate ur enthusiasm but as the title clearly says, its a challenge for pardesi :D

nevertheless pure optics is enough :P

A stone lies on the bottom of a stream.A boy wants to hit it with a stick .Taking aim the boy holds the stick in the air at an angle $rac {pi}{4}$ with the horzontal.At what distance from the stone will the stick hit the bottom if the depth of the lake is $32$ cm.

pardesi its a challenge to you! Try it da :P
Why pardesi? Bcos he's one of the best phy problem solvers here :P

oxdn state is a hypothetical concept started to understand things better. Hence we cannot explain the 0 oxdn state of O in HOF.

hey pardesi alais mr K V shuddhodan why face not seen @ aops? Busy nving fer the boards? And serious fer iit?

well there is an explanation there!
Well you can represent a + b = c, a , b >= 0 as also the coefficient of x^c in the expansion of (1 + x + x^2........+x^c)(1+x + x^2 + .... + x^c) becos we notice that if ax^m and bx^n multiply to give ab x^(m+n) and m+n = c, then m, n is a soln of the eqn a + b = c :)

wat do u mean by that computer001?

wats that?

No, learning them is very easy using kiliani fischer synthesis just remember them using their first letters :D