There should a limit somewhere

Calculus Level 2

True or False?

1 + 2 + 3 + 4 + 5 + 6 + 2 + 4 + 6 + 8 + 10 + 12 + = 1 2 \dfrac{1 + 2 + 3 +4 + 5 + 6+ \cdots } {2 + 4 + 6 + 8 + 10 + 12 + \cdots } = \dfrac12

True False

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1 solution

Zach Abueg
Jan 17, 2017

Both series diverge to \displaystyle \infty , so the fraction is indeterminate: \displaystyle \frac {\infty}{\infty} .

Applying a limit, however, would give us 1 2 \displaystyle \frac 12 :

lim n ( n ) ( n + 1 ) 2 ( n ) ( n + 1 ) \displaystyle \lim_{n \to\infty} \frac {\frac {(n)(n + 1)}{2}}{(n)(n + 1)} = lim n ( n ) ( n + 1 ) 2 ( n ) ( n + 1 ) = \displaystyle = \lim_{n \to\infty} \frac {(n)(n + 1)}{2(n)(n + 1)} = \frac {\infty}{\infty}

Apply L'hopital's Rule:

lim n n 2 + n 2 n 2 + 2 n = lim n 2 n + 1 4 n + 2 = lim n 2 4 = lim n 1 2 = 1 2 \displaystyle \lim_{n \to\infty} \frac {n^2 + n}{2n^2 + 2n} = \lim_{n \to\infty} \frac {2n + 1}{4n + 2} = \lim_{n \to\infty} \frac 24 = \lim_{n \to\infty} \frac 12 = \frac 12

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 = k = 1 n k 2 k = 1 n + 1 k a_n = \dfrac{\sum_{k=1}^n k} {2\sum_{k=1}^{n+1} k} which obviously has limit 1 2 \dfrac{1}{2} while for all n N + n \in \mathbb N_+ the fraction lies below. So 'true' is actually the right answer for this choice of enumeration.

Marcel Stevee - 4 years, 4 months ago

Why is 2^-1 wrong answer when all of you say is the correct

Said Pattinson - 3 years, 5 months ago

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They didn't ask for limit

Shiv Kiran Bagathi - 11 months ago

Why did you use l'hospital's rule for infinity/infinity? isn't it just for 0/0.

imad hamaidi - 2 years, 4 months ago

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Infinity/infinity can also be written as ( 1 by 0)/(1 by 0) so it becomes 0/0 form

Srinivas Dixit - 2 years, 4 months ago

Is the statement true or false? What is the correct answer?

Tripty Dubey - 2 years, 2 months ago

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.

Aayush Gupta - 1 year, 2 months ago

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