# average value, functions

#### idontknow

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
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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idontknow

#### idontknow

I agree that (2) seems non-sense to be 0 since $$\displaystyle \dfrac{1}{n}$$ cannot be 0.

#### romsek

Math Team
(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

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

idontknow

#### mathman

Forum Staff
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.

idontknow

#### Maschke

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.

idontknow