Recent content by topengonzo

I just have to prove ab + 2/ab -4 >= 25/2 when a+b=1 How to do it without Lagrange

the equations are: 2(a^4) - λ (a^3) -2 = 0 2(b^4) - λ (b^3) -2 = 0 a+b-1=0 I can solve by trial and error to get a=b=0.5 but what if there is another root for these equations

I differentiated it with respect to a and i get the equation a - 1/(a^3) - (1-a) + 1/(1-a)^3 = 0 which I can't solve :S

Homework Statement How to prove (a+1/a)^2 + (b+1/b)^2 >= 25/2 given that a+b=1 and a,b positive Homework Equations The Attempt at a Solution I tried replacing b with 1-a but I get 6th degree a which I don't know how to find inequality for. Since a and b are perfect symmetric in...

Shouldn't I check at x=y and also at the boundaries that is x=0,y=10 and x=10,y=0?

How to prove it?

Your right. Rereading the lecture, they mentioned that that a1,..,an and b1,...,bn are in descending order and they are all positive.

Is my solution correct and is there another way to solve it?

\sum_{r=k+1}^{n+1} \frac {1} {(r-1)!} - \sum_{r=k+1}^{n+1} \frac {1} {(r)!} YOUR RIGHT! it becomes simply \frac {1} {(k)!} - \frac {1} {(n+1)!} Just 1 final question? I take lim as n tends to infinity of my answer so I get that the infinite sum of all terms after ak/k! is...

\sum_{r=k+1}^{n+1} \frac {r-1} {(r)!} < \frac 1 {k!} Is this correct? \sum_{r=0}^{inf} \frac {r-1} {(r)!} = e-e=0 And then I take out term from 0 to k?

I think that your previous answer using series will solve it. I am thinking of putting \sum_{r=k}^n \frac r {(r+1)!} < \frac 1 {k!} and I use the poisson distribution series to prove = 1/k! and thus the sum of any finite terms of the series is < 1/k! Am I correct?

As I can see from the formula of cauchy inequality: (a1^2+a2^2+...+an^2)^1/2 . (b1^2+b2^2+...+bn)^1/2 >= a1b1+a2b2 + ... + anbn Can I conclude from the above formula that: (a1+a2+...+an)^1/2 . (b1+b2+...+bn)^1/2 >= (a1b1)^1/2 + (a2b2)^1/2 +...+ (anbn)^1/2 by setting a1,...,an =...

This is not the question of my homework. I will write it here down to show you that I am doing an effort and what I am asking will lead me to the answer. Question: Prove every positive rational number x can be expressed in ONE way in the form x= a1 + a2/2! + a3/3! + ... + ak/k! where...

Yes the second one you wrote is what i exactly mean. How do I prove it? Also I think if i set n -> infinity (find lim at infinity), I would get = instead of < . Am I correct?

n is induction variable and base term is n=k