Riemann's paradox: pi = infinity minus infinity
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it when explaining why conditionally convergent series can change their sum when rearranged and why absolute convergence is safer.
Recommended by 3 people across 3 different months, 2016 to 2021.
What it is
Mathologer explains Riemann's rearrangement theorem, showing how the terms of the conditionally convergent alternating harmonic series can be reordered to sum to any value, including pi. The animated explainer frames this as subtracting infinity from infinity in a legitimate way.
What people said
Strong rec Sean Lake · 2016-09 · Mathematics Stack Exchange · on “What's the significance of conditional and absolute convergence?”
There's a theorem that proves that if a series is conditionally convergent then it is possible to rearrange the terms of the sum so that it converges to any value. Basically, if you sum the positive and negative terms separately you'll get $\infty - \infty$, an indeterminate form that depends on how the two terms are diverging. The bottom line is that an absolutely convergent series is more robust to manipulations without having to worry about whether the answer is altered by them. Mathologer has a good video on this aspect of conditional convergence [this video] .
Recommends Crostul · 2017-02 · Mathematics Stack Exchange · on “Rearrangements of the harmonic series”
This video may be instructive [this video] . Anyway, there is no trivial reason to conclude that such an arrangment exists. You can arrange the summand in order to get that the infinite series is $0$, but this does not mean that you can find some arrangement where the first 10 terms add up to $0$.
Recommends Ski Mask · 2021-01 · Mathematics Stack Exchange · on “Understanding how different groupings of terms in an Infinite series can lead to different answers.”
One of the most, conceptually speaking, for me to understand is the topic of infinite series. I have always had a hard time proving that an infinite series diverges or even finding a solution for the sum of infinite series. So when I came across a video by Mathologer titled " Riemann's paradox: pi = infinity minus infinity [this video] ", my mind had a mini-meltdown. In the video he mentions Reimann's Rearrangement Theorem, whereby the sum of the infinite series $1- \frac{1}{2}+\frac{1}{1}-\frac{1}{4}+...$ can equal whatever you want it to, depending on how you group the terms. Even though in the video he explains how that is possible, I cannot for the life of me understand how different grouping
Cites u8y7541 · 2017-06 · Mathematics Stack Exchange · on “Question about validity of the Maclaurin Series proof of Euler's identity”
I recently watched a YouTube video illustrating that using commutativity may change the value of an infinite series (Riemann's paradox, [this video] ). The identity $e^{i\theta} = \cos\theta + i\sin\theta$ is often proved by taking the infinite Maclaurin series of $\cos\theta$ and $i\sin\theta$ and rearranging the terms to show that the resulting series is equal to the Maclaurin series of $e^{i\theta}$. However, I am now in doubt of the validity of this proof because commutativity may not be valid when dealing with infinite series. Can someone explain why this proof is still valid?