n → ∞ lim j = 1 ∑ n 1 ⎝ ⎛ j = 1 ∑ n 2 j ⎠ ⎞ − ⎝ ⎛ j = 1 ∑ n 1 ⎠ ⎞ 2 = ?
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What a laughter! I up voted. The question does mean that each series are having exactly same total number of terms. However, strict staff shall oppose to you I think.
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Yes, my initial intention was like that, but I've already changed it to a limit form to avoid infinity dilemma.
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Now they share exactly the same description at once. However, many people shall find this easy because of the change of description. Hard to make it fantastic and also complying to mathematics wanted.
Guys!
Sum of natural numbers till 'n' = n(n+1)÷2
Here:
2+4+6+8... = 2(1+2+3+4...)
So, n(n+1)÷2 = 2(1+2+3+4...)
=> n(n+1) = 1+2+3+4...
Now we can say that n = infinity as the series does not end. So, n=1+1+1+1... => taken alphabetically, [{n(n+1)}÷n] - n => {(n²+n)-n²}÷n = n÷n = 1
I think this is the right method to solve the question. Any corrections are welcomed.
Why (1+1+1..……)=n and why there is n in(n(n+1)/2)?
But how? What does n stand for?
= n → ∞ lim j = 1 ∑ n 1 ⎝ ⎛ j = 1 ∑ n 2 j ⎠ ⎞ − ⎝ ⎛ j = 1 ∑ n 1 ⎠ ⎞ 2 = n 2 2 n ( n + 1 ) − n 2
= n + 1 − n = 1
When you see a problem that looks extremely hard but its like 30 points say 1.
I don't know sign mean but its like a simple algebraic expression j=1
then 2j =2
so it will be (2-1)/1=1
Wrong .... Not clear
Yaa I don't know sign before but I know now
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Using the sum of the first n natural numbers formula ,
( n 2 2 n ( n + 1 ) ) − n
n n ( n + 1 ) − n
n + 1 − n = 1