average value, functions

Dec 2015
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Find the average value of the functions below :
(1) \(\displaystyle f(x)=\lfloor x \rfloor ^x \; , \: x\in [1,2]\in \mathbb{R}\).
(2) \(\displaystyle a_n =\dfrac{1}{n} \; , \: n\in \mathbb{N} \).
Method required !
 

romsek

Math Team
Sep 2015
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Well (1) is obviously 1 as the measure of the set where $\lfloor x \rfloor = 1, ~x \in [1,2]$ is $1$ and $1^x = 1~ \forall x$
The only other value $f(x)$ has on this interval is $f(2)=4$, but the interval $[2,2]$ has measure 0 so adds nothing to the average.

(2) $\bar{a} = \lim \limits_{N\to \infty} \sum \limits_{n=1}^N \dfrac{1}{n} =

\lim \limits_{N\to \infty} \dfrac 1 n H(n) \leq \lim \limits_{N\to \infty} \dfrac{\ln(N)+1}{N} = 0$

To be honest, I find the result in (2) troubling. The infinite sum of positive values produces an average of 0. That doesn't make sense to me, but such is the nature of infinity.
 
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Dec 2015
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I agree that (2) seems non-sense to be 0 since \(\displaystyle \dfrac{1}{n}\) cannot be 0.
 

romsek

Math Team
Sep 2015
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(2) $\bar{a} = \lim \limits_{N\to \infty} \sum \limits_{n=1}^N \dfrac{1}{n} =

\lim \limits_{N\to \infty} \dfrac 1 n H(n) \leq \lim \limits_{N\to \infty} \dfrac{\ln(N)+1}{N} = 0$

To be honest I find the result in (2) troubling. The infinite sum of positive values produces an average of 0. That doesn't make sense to me but such is the nature of infinity.
those equations above are a mess. They should read

$\bar{a} = \lim \limits_{N\to \infty} \dfrac 1 N \sum \limits_{n=1}^N \dfrac 1 n = \lim \limits_{N\to \infty} \dfrac 1 N H_N \leq \lim \limits_{N\to \infty}\dfrac{\ln(N)+1}{N} = 0$
 
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mathman

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May 2007
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I agree that (2) seems non-sense to be 0 since \(\displaystyle \dfrac{1}{n}\) cannot be 0.
It seems almost logical. However the limit is less than any specified positive value, so it must be zero.
 
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Aug 2012
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To be honest I find the result in (2) troubling. The infinite sum of positive values produces an average of 0. That doesn't make sense to me but such is the nature of infinity.
The tail keeps pulling the average down. Sort of makes sense.
 
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