Friday, 12 August 2016

real analysis - Finding limxtoinftyleft(fraca1/x1+a1/x2+cdotcdotcdotan1/xnright)nx



For non zero positive reals a1,a2an how to find
limx(a1/x1+a1/x2++a1/xnn)nx?




It becomes indeterminant form 1^{\infty}. But difficult to solve by L'Hospital's Rule. By using A.M.-G.M. inequality it comes that limit is \geq a_{1}a_{2}\cdot\cdot\cdot a_{n}. I also tried by using Squeeze. Please help.Thanks.


Answer



tl;dr




For a_1,\dots, a_n positive numbers, and x>0, \lim_{x\to\infty}\left(\frac{a_{1}^{\frac{1}{x}}+a_{2}^{\frac{1}{x}}+\cdots+{a_{n}}^{\frac{1}{x}}}{n}\right)^{nx} = \prod_{k=1}^na_{k}





Proof. Detailed approach, where we use the (low-order) Taylor expansions e^u=1+u+o(u) and \ln(1+u)=u+o(u) when u\to 0.



You can rewrite
\begin{align} \left(\frac{a_{1}^{\frac{1}{x}}+a_{2}^{\frac{1}{x}}+\cdots+{a_{n}}^{\frac{1}{x}}}{n}\right)^{nx} &= \exp\left(nx \ln \frac{a_{1}^{\frac{1}{x}}+a_{2}^{\frac{1}{x}}+\cdots+{a_{n}}^{\frac{1}{x}}}{n}\right) = \exp\left(nx \ln \frac{1}{n}\sum_{k=1}^na_{k}^{\frac{1}{x}}\right)\\ &= \exp\left(nx \ln \frac{1}{n}\sum_{k=1}^n e^{\frac{1}{x}\ln a_{k}}\right) \end{align}
Since n is a constant and \frac{1}{x}\ln a_{k}\xrightarrow[x\to\infty]{} 0 for each k, we have
e^{\frac{1}{x}\ln a_{k}} = 1 + \frac{1}{x}\ln a_{k} + o\left(\frac{1}{x}\right)
for each k as x\to\infty, and therefore
\begin{align} \left(\frac{a_{1}^{\frac{1}{x}}+a_{2}^{\frac{1}{x}}+\cdots+{a_{n}}^{\frac{1}{x}}}{n}\right)^{nx} &= \exp\left(nx \ln \frac{1}{n}\sum_{k=1}^n e^{\frac{1}{x}\ln a_{k}}\right)\\ &= \exp\left(nx \ln\left( 1+ \frac{1}{nx}\sum_{k=1}^n\ln a_{k}+ o\left(\frac{1}{x}\right)\right)\right) \\ &= \exp\left(nx \left( \frac{1}{nx}\sum_{k=1}^n\ln a_{k}+ o\left(\frac{1}{x}\right)\right)\right) \\ &= \exp\left(\sum_{k=1}^n\ln a_{k}+ o\left(1\right)\right) \\ &= \exp\left(\ln \prod_{k=1}^na_{k}+ o\left(1\right)\right) = e^{o(1)} \prod_{k=1}^na_{k} \\ & \xrightarrow[x\to\infty]{} \prod_{k=1}^na_{k}. \end{align}


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