Hi everybody,
I got severely stuck with this. Could someone tell me how to approach this?

Suppose is a sequence of +ve real nos. such that x₁ > = x₂ > = ...> = x_n ... for all n and for all n
x₁ /1 + x_2^2 /2 + ... + x_n^2 /n > = 1
Show that for all k the following inequality is satisfied-
x₁ /1 + x₂ /2 + x₃ /3 +... + x_k /k > = 3.
[here xi denotes x with a subscript i]
Regards,
Archie.

Your fourth line reads:

x_1^2 /1 + x_2^2 /2 + ... + x_n^2 /n < = 1

We want:

x₁ /1 + x₂ /2 + ... + x_n /n < = 3

We will actually show that for each n:

x₁ /1 + x₂ /2 + ... + x_n^2-1 /(n² -1) < = 3

Can you see why this is sufficient to prove the entire claim?

Anyway, we have:

x₁ /1 + ... + x_n^2-1 /(n² - 1)

= x₁ /1 + x₂ /2 + x₃ /3
+ x₄ /4 + x₅ /5 + x₆ /6 + x₇ /7 + x₈ /8
+ x₉ /9 + x₁₀ /10 + ... + x₁₅ /15
+ ...
+ x_(n-1)^2 /(n-1)^2 + x_(n-1)^2+1 /((n-1)^2+1) + ... + x_n^2-1 /(n² -1)

(so we've separated the terms into lines, each line starting with a square).

Because x_i < = x_i+1 this is expression is:

< = x₁ (1/1 + 1/2 + 1/3)
+ x₄ (1/4 + 1/5 + ... + 1/8)
+ x₉ (1/9 + 1/10 + ... + 1/15)
+ ...
+ x_(n-1)^2 (1/(n-1)² + 1/((n-1)² +1) + ... + 1/(n² -1))

Now notice that the right hand bracket on the ith line is 1/i² + 1/(i² + 1) + ... + 1/((i+1)² -1). Can you see why this is always less than or equal to 3? If so you can see that the original expression is smaller than or equal to:

3x₁ + 3x₄ + 3x₉ + ... + 3x_(n-1)^2

which is less than or equal to 3. So we're done.

Hi,
I couldnt uderstand why is it sufficient to show that x₁ /1 + x₂ /2 + ... + x_n^2-1 /n² -1 > = 3 to prove the whole thing. Could you explain it again?

Apologies, I somehow managed to miss your message. It's probably now too late, but anyway the reason is that more terms you add, the larger the sum gets. So if you want to show that the sum is always less than 3 after 10 terms, it suffices to prove that the sum is less than 3 after (say) 100 terms. In other words you only have to prove your conjecture for arbitrarily large n. So if you can verify your conjecture for n = 1² , 2² , 3² , ... then it must certainly hold for all n.