The bizarre world of INFINITE rearrangements // Riemann Series Theorem

We can rearrange a finite sum like 1+2+3=3+2+1, no big deal. But what if we make infinite rearrangements on an infinite sum? We're going to see that we can actually use this to add up to any number you like! The key idea is about conditionally convergent series, that is a series which converges because of lots of positives and negatives cancelling, but does not converge if you take the absolute value of each term. Then the Reimann Rearrangement Theorem tells us such conditionally convergent series can always have a rearrangement to add up to any given value, or even diverge to infinity. If the series absolutely converges, then things are much nicer and you can rearrange without changing the value it converges. We're going to see two specific series in this video. The Geometric Series 1/2+1/4+1/8+1/16+... is a convergent series and we will argue what it means that such a series should indeed converge. Then we will talk about the Alternating Harmonic Series 1-1/2+1/3-1/4+1/5-1/6+... which is conditionally convergent and I"ll show how we can rearrange this to sum to either ln(2) or 1/2 ln(2)!! Proof without words that the Alternating Harmonic Series sums to ln(2): https://www.maa.org/sites/default/files/Hudleson-MMz-201007804.pdf Divergence of the Harmonic Series:https://www.youtube.com/watch?v=5ejmgwXVSqQ 0:00 Finite Rearrangements 1:00 Convergence of Geometric Series 4:30 Rearranging the Alternating Harmonic Series 9:50 Reimann Series Theorem **************************************************** COURSE PLAYLISTS: ►CALCULUS I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m ► CALCULUS II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc4ySKTIW19TLrT91Ik9M4n ►MULTIVARIABLE CALCULUS (Calc III): https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc_CvEy7xBKRQr6I214QJcd ►DIFFERENTIAL EQUATIONS (Calc IV): https://www.youtube.com/watch?v=xeeM3TT4Zgg&list=PLHXZ9OQGMqxcJXnLr08cyNaup4RDsbAl1 ►DISCRETE MATH: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS ►LINEAR ALGEBRA: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6 *************************************************** ► Want to learn math effectively? Check out my "Learning Math" Series: https://www.youtube.com/watch?v=LPH2lqis3D0&list=PLHXZ9OQGMqxfSkRtlL5KPq6JqMNTh_MBw ►Want some cool math? Check out my "Cool Math" Series: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho **************************************************** ►Follow me on Twitter: http://twitter.com/treforbazett ***************************************************** This video was created by Dr. Trefor Bazett. I'm an Assistant Teaching Professor at the University of Victoria. BECOME A MEMBER: ►Join: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join MATH BOOKS & MERCH I LOVE: ► My Amazon Affiliate Shop: https://www.amazon.com/shop/treforbazett