Exponentiation Question

What is limnAB,\displaystyle\lim_{n\rightarrow\infty}\dfrac{A}{B}, where A=n(n1)(n2)32A=n^{(n-1)^{(n-2)^{\ldots^{3^2}}}} and B=234(n1)nB=2^{3^{4^{\ldots^{(n-1)^n}}}}? I suspect it is 0,0, but I don't have any idea how I would go about proving this. A little help?

Note by Trevor B.
4 years, 6 months ago

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1 vote

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suppose the value of pie =0.31825. calculate the area of circle of radius= 5 equal to 78.55 . please share the answer ok.

amar nath - 4 years, 6 months ago

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I have a question, though. If the B grows faster, than the equation would near zero. If A grows faster, the equation would near infinity... right?

Am I missing something right now?

CS ಠ_ಠ Lee - 4 years, 6 months ago

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Wait, I think im pretty close.

CS ಠ_ಠ Lee - 4 years, 6 months ago

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The way I look at it, the limit of AB\frac{A}{B} would be in the form infinf\frac{\inf}{\inf},

CS ಠ_ಠ Lee - 4 years, 6 months ago

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Certainly, but the question would be which sequence (see Calvin Lin's comment) grows faster.

Jake Lai - 4 years, 6 months ago

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Oh, ok, thanks

CS ಠ_ಠ Lee - 4 years, 6 months ago

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Those terms should be defined as An A_n and BnB_n.

Hint: How does An A_n and An1 A_{n-1} relate? What about Bn B_n and Bn1 B _{n-1} ?

Which test does that suggest we apply?

Calvin Lin Staff - 4 years, 6 months ago

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While it is obvious that An=nAn1,A_n=n^{A_{n-1}}, I can't come up with a mathematical relartion for Bn1B_{n-1} and Bn,B_n, since BnBn1n.B_n\neq B_{n-1}^n.

Trevor B. - 4 years, 6 months ago

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