True or False?
2 + 4 + 6 + 8 + 1 0 + 1 2 + ⋯ 1 + 2 + 3 + 4 + 5 + 6 + ⋯ = 2 1
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Since both the enumerator and denominator are divergent series, the answer depends on how the sequence is counted.
In the picture we have: a n = 2 ∑ k = 1 n + 1 k ∑ k = 1 n k which obviously has limit 2 1 while for all n ∈ N + the fraction lies below. So 'true' is actually the right answer for this choice of enumeration.
Why is 2^-1 wrong answer when all of you say is the correct
Why did you use l'hospital's rule for infinity/infinity? isn't it just for 0/0.
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Infinity/infinity can also be written as ( 1 by 0)/(1 by 0) so it becomes 0/0 form
Is the statement true or false? What is the correct answer?
It is not written then both the sums are adding upto n. As both are tending to infinity it might be the case one of them is being summed upto n^2 or has many other possibilties and limit won't come as 0.5. So we can't say that the answer is 0.5.
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Both series diverge to ∞ , so the fraction is indeterminate: ∞ ∞ .
Applying a limit, however, would give us 2 1 :
n → ∞ lim ( n ) ( n + 1 ) 2 ( n ) ( n + 1 ) = n → ∞ lim 2 ( n ) ( n + 1 ) ( n ) ( n + 1 ) = ∞ ∞
Apply L'hopital's Rule:
n → ∞ lim 2 n 2 + 2 n n 2 + n = n → ∞ lim 4 n + 2 2 n + 1 = n → ∞ lim 4 2 = n → ∞ lim 2 1 = 2 1