i somehow managed to prove the opposite :/.plz tell my mistakea(n) = f(1) + f(2) + ... +f(n) - 1 ∫ n f(x)dxdifferentiating wrt to na'(n)= -f(n)again differentiating wrt to na"(n)= -f'(n)f'(n)<0⇒ a"(n)>0 (d^2a/dn^2 >0,concavity)⇒ a(n)is concave for all nif it always remains concave,its graph will diverge.
write x^2 as x*x=x+x+x+x+x+x....(x times)differentiating wrt x..2x=1+1+1+1+1+1+1+1....(x times)=>2x=xabsurd, isn't it?if you get your mistake from this, do tell.
where i see the mistake is in this thing-2x=1+1+1+1+1+1+1+1....(x times)if we put x a non intger.say 1.5,LHS=3 but we can't add 1,1.5 times.so i think the above equation is true ONLY for integers?
ok.got it now.
i somehow managed to prove the opposite :/.plz tell my mistake
ReplyDeletea(n) = f(1) + f(2) + ... +f(n) - 1 ∫ n f(x)dx
differentiating wrt to n
a'(n)= -f(n)
again differentiating wrt to n
a"(n)= -f'(n)
f'(n)<0
⇒ a"(n)>0 (d^2a/dn^2 >0,concavity)
⇒ a(n)is concave for all n
if it always remains concave,its graph will diverge.
write x^2 as x*x=x+x+x+x+x+x....(x times)
ReplyDeletedifferentiating wrt x..
2x=1+1+1+1+1+1+1+1....(x times)
=>2x=x
absurd, isn't it?
if you get your mistake from this, do tell.
where i see the mistake is in this thing-
ReplyDelete2x=1+1+1+1+1+1+1+1....(x times)
if we put x a non intger.say 1.5,LHS=3 but we can't add 1,1.5 times.so i think the above equation is true ONLY for integers?
ok.got it now.
ReplyDelete