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$V_a^b(P,f):=\sum_{i=1}^k|f(x_i)-f(x_{i-1})|$, where $P$ is a partition.

$f(x)= \begin{cases} x^2\sin(\frac{1}{x^2}), &\text{if } x\neq0, \\ 0, &\text{if } x=0 \end{cases} $ does not have bounded variation over $[-1,1]$. I'm trying to show this without using the fact that it may not have bounded variation over some subset of $[-1,1]$ .

I want to find a partition $\lbrace x_0,\dotsc,x_n \rbrace$ of $[-1,1]$ for which $\sum_{i=1}^n|f(x_i)-f(x_{i-1})|$ gives a partial sum of the Harmonic Series. My partition is $P=\lbrace -1,0,\frac{1}{\sqrt{\frac{\pi}{2}+\pi n}},\frac{1}{\sqrt{\frac{\pi}{2}+\pi(n-1)}},\dotsc,\frac{1}{\sqrt{\frac{\pi}{2}}},1\rbrace$.

$\begin{align} V_{-1} ^1(P,f) &=|f(0)-f(-1)| + \left|(f\Bigl(\frac1{\sqrt{\smash[b]{\frac\pi2}+\pi n}}\Bigr)-f(0)\right|+\left|f(1)-f\Bigl(\frac1{\sqrt{\smash[b]{\frac\pi2}}}\Bigr)\right| \\ &\mathrel{\phantom=} +\sum_{i=0}^{n}\left|f\Bigl(\frac1{\sqrt{\smash[b]{\frac\pi2}+\pi (n+1)}}\Bigr)-f\Bigl(\frac1{\sqrt{\smash[b]{\frac\pi2}+\pi n}}\Bigr)\right| \\ &= |f(0)-f(-1)|+\left|f\Bigl(\frac1{\sqrt{\smash[b]{\frac\pi2}+\pi n}}\Bigr)-f(0)\right| \\ &\mathrel{\phantom=}+\left|f(1)-f\Bigl(\frac1{\sqrt{\smash[b]{\frac\pi2}}}\Bigr)\right|+\sum_{i=0}^n\left(\frac1{\smash[b]{\frac\pi2}+\pi n}+\frac1{\smash[b]{\frac\pi2}+\pi (n-1)}\right). \end{align}$

I don't know what to do next with the summation. Am I on to something?

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    That is nice. Didn't know I can do that.2012-11-23

1 Answers 1

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I think the problem is that your sum is not right. It should be $ \begin{align} \sum_{k=0}^n\left(\frac1{\frac\pi2+\pi(k+1)}+\frac1{\frac\pi2+\pi k}\right) &=\frac2\pi+\frac2\pi\sum_{k=1}^n\frac1{k+\frac12}\\ &\ge\frac2\pi+\frac2\pi\sum_{k=1}^n\frac1{k+1}\\ &=\frac2\pi\sum_{k=1}^n\frac1k \end{align} $