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An ISI 2010 Problem

Let \(a_1>a_2>.......>a_r\) be positive real numbers. Compute \(\lim_{n \to \infty} ({a_1}^n +{a_2}^n + ........+ {a_r}^n)^{\frac{1}{n}}\).

[Please be kind enough to write the solution.]

Note by Maharnab Mitra
3 years, 2 months ago

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Let \[L = \lim_{x \to \infty} (a_1^x + a_2^x + \dots + a_r^x)^{1/x}.\] Then \[ \begin{align*} \log L &= \log \lim_{x \to \infty} (a_1^x + a_2^x + \dots + a_r^x)^{1/x} \\ &= \lim_{x \to \infty} \log (a_1^x + a_2^x + \dots + a_r^x)^{1/x} \\ &= \lim_{x \to \infty} \frac{\log (a_1^x + a_2^x + \dots + a_r^x)}{x}. \end{align*} \]

By L'Hopital's Rule, \[ \begin{align*} \lim_{x \to \infty} \frac{\log (a_1^x + a_2^x + \dots + a_r^x)}{x} &= \lim_{x \to \infty} \frac{\frac{d}{dx} \log (a_1^x + a_2^x + \dots + a_r^x)}{\frac{d}{dx} x} \\ &= \lim_{x \to \infty} \frac{a_1^x \log a_1 + a_2^x \log a_2 + \dots + a_r^x \log a_r}{a_1^x + a_2^x + \dots + a_r^x}. \end{align*} \]

Then \[ \begin{align*} \lim_{x \to \infty} \frac{a_1^x \log a_1 + a_2^x \log a_2 + \dots + a_r^x \log a_r}{a_1^x + a_2^x + \dots + a_r^x} &= \lim_{x \to \infty} \frac{(a_1^x \log a_1 + a_2^x \log a_2 + \dots + a_r^x \log a_r)/a_1^x}{(a_1^x + a_2^x + \dots + a_r^x)/a_1^x} \\ &= \lim_{x \to \infty} \frac{\log a_1 + (a_2/a_1)^x \log a_2 + \dots + (a_r/a_1)^x \log a_r}{1 + (a_2/a_1)^x + \dots + (a_r/a_1)^x} \\ &= \log a_1. \end{align*} \]

Therefore, \(L = a_1\). Jon Haussmann · 3 years, 2 months ago

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@Jon Haussmann Thanks :) Maharnab Mitra · 3 years, 2 months ago

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Take a1 to power n common from bracket..it become lim(a1)[1 + (a2/a1)^n +....]^1/n .

It all terms terms become zero ..so answer will come to ...a1. Aman Chaudhary · 3 years, 2 months ago

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@Aman Chaudhary Not immediately true. The limiting value of \( 0 ^ 0 \) depends on what path you take.

Edit: Ooops, I spoke too quickly. The power needs to be logarithmic. Use \( f(x) = x, g(x) = \frac{ 1} {\log x} \), then we have \( f(x) ^ { g(x) } = e \). Calvin Lin Staff · 3 years, 2 months ago

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@Calvin Lin \(\lim_{x \to 0} x^{x^0.2} = e^{\lim_{x \to 0} x^{\frac{1}{5}} ln (x)}\)

Now, \(\lim_{x \to 0} x^{\frac{1}{5}} ln (x)= \lim_{x \to 0} \frac{ ln (x)}{x^{\frac{-1}{5}}}\) \(=\lim_{x \to 0} \frac{-x^{\frac{1}{5}}}{5}\) (using \(L'Hospital\)'s rule) which goes to 0. Thus, the required limit is \(e^0=1\) Maharnab Mitra · 3 years, 2 months ago

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@Maharnab Mitra Sorry, updated. Calvin Lin Staff · 3 years, 2 months ago

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@Calvin Lin Sir, then what should the correct answer (or rather solution) to the problem be? Maharnab Mitra · 3 years, 2 months ago

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Multiply and divide by a1 and proceed. Ans: a1 Pritiranjan Dwibedy · 3 years, 1 month ago

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