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Alternating harmonic series, $\frac11-\frac12+\frac13-\frac14+\frac15-\frac16+ \ldots$ converges to $\log(2)$,

and the rearranged series, $\frac11-\frac12-\frac14+\frac13-\frac16-\frac18+\frac15-\frac{1}{10}-\frac{1}{12} +\ldots$ converges to $\frac{\log(2)}{2}$.

Since the rearrangement is a bijective function, it contradicts with my intuition to have different value when summed.

Therefore, I tried to understand this situation, treating infinity as the end ( taking the sum on the interval $[0, ∞]$ instead of $[1,∞)$. )

Then, the original series becomes: $\frac11-\frac12+\frac13-\frac14+...+\frac1∞-\frac1∞$

and the rearranged series becomes: $\frac11-\frac12-\frac14+\frac13-\ldots+\frac{1}{(2∞/3)}-\frac{1}{(4∞/3)}$

Taking the difference of sums: $\frac{1}{(2∞/3)} + \ldots + \frac1∞ + \ldots + \frac{1}{(4∞/3)}$

This equals to $\int_0^1\frac{1}{(2/3 + 2x)}$ dx which is $\frac{\log(2)}{2}$.

I wonder if this way of thinking is allowed or if it reveals my lack of understanding of infinity.

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    $\begingroup$ Welcome to MSE. Please have a look at this helpful MathJax reference and guide to learn how to typeset mathematical expressions. $\endgroup$ – Théophile Apr 12 '19 at 21:11
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    $\begingroup$ If you want to work with infinity, use the Projective Real line, or append the one point $\infty$ to $\mathbb{R}_+.$ You cannot subtract infinities, though in the right contexts (e.g. measure theory) you might define $0 \cdot \infty = 0$ to make arguments simpler. In general, you either need to carefully define your number system or instead work with limiting behavior. You are being too cavalier and making amateur mistakes; this is not unexpected from someone new to working with extended reals, but err on the side of caution for now. $\endgroup$ – Brevan Ellefsen Apr 12 '19 at 21:14
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The Riemann series theorem shows that the terms of a conditionally convergent series can be rearranged to converge to any value you like. This would be a good place to start.

In general, unless you really know what you're doing, you never want to manipulate $\infty$ as if it had some value. When people write $1+\frac12+\frac13+\cdots=\infty$, for example, this is really just a shorthand for a limit: $$\lim_{n \to \infty}1+\frac12+\frac13+\cdots+\frac1n=\infty,$$ and even then, the "equality" is a shorthand for indicating that the left-hand side grows without bound. With that in mind, it doesn't make sense to subtract $\infty$ from both sides.

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  • $\begingroup$ so the way I interpreted the situation is a bad and wrong interpretation? $\endgroup$ – mathlearner98 Apr 12 '19 at 21:36
  • $\begingroup$ I tried to understand this situation as: the original series and the rearranged series share all the elements that have a fraction with natural number, but do not share infinitesimal, which cause the difference. $\endgroup$ – mathlearner98 Apr 12 '19 at 21:41
  • $\begingroup$ @phyam What do you mean exactly by infinitesimals? The terms consist only of fractions. $\endgroup$ – Théophile Apr 12 '19 at 21:55

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