--- title: "Meaning & Key Takeaway of: Notable enough, however, are the…" description: "\"Notable enough, however, are the controversies over the series 1 - 1 + 1 - 1 + 1 - ... whose sum was given by Leibniz as 1/2, although others disagree...." image: "https://sms4smile.com/quotes/i/7a5/leonhard-euler-28718_controversy_notable-enough-however-are-the-controver_3187478.jpg" --- # Notable enough, however, are the controversies over the… ![“Notable enough, however, are the controversies over the series 1 - 1 + 1 - 1 + 1 - ... whose sum was given by Leibniz as 1/2, although others disagree. ... Understanding of this question is to be sought in the word "sum"; this idea, if thus conceived - namely, the sum of a series is said to be…” quote by Leonhard Euler](https://sms4smile.com/quotes/i/7a5/leonhard-euler-28718_controversy_notable-enough-however-are-the-controver_3187478.jpg) [Download](https://sms4smile.com/quotes/i/7a5/leonhard-euler-28718_controversy_notable-enough-however-are-the-controver_3187478.jpg) [Open image](https://sms4smile.com/quotes/i/7a5/leonhard-euler-28718_controversy_notable-enough-however-are-the-controver_3187478.jpg) > “Notable enough, however, are the controversies over the series 1 - 1 + 1 - 1 + 1 - ... whose sum was given by Leibniz as 1/2, although others disagree. ... Understanding of this question is to be sought in the word "sum"; this idea, if thus conceived - namely, the sum of a series is said to be that quantity to which it is brought closer as more terms of the series are taken - has relevance only for convergent series, and we should in general give up the idea of sum for divergent series.” [Leonhard Euler](https://sms4smile.com/quotes/author/leonhard-euler/) ## About This Quote This interpretation was drafted with AI assistance. It is one reading of the quote, not the author's own explanation. Leibniz’s series sum 1/2 applies only to convergent series; divergent series lack a true sum. **In simple terms:** Only convergent series have a definite sum. Key Takeaway Treat divergent series differently. Themes mathematics series convergence Mood analytical inquisitive Type educational philosophical When to use this quote - calculus teaching - advanced mathematics research - theoretical physics - numerical methods Key Concepts analysis limit divergence Questions to Reflect On - How do we handle divergent series in practice? - What alternative summation techniques exist? A Different Perspective Divergent series can be assigned summation methods, but they differ from ordinary sums. ★ ★ ★ ★ ★ No ratings yet [Controversy](https://sms4smile.com/quotes/about/controversy/) [Divergent series](https://sms4smile.com/quotes/about/divergent-series/) [Education](https://sms4smile.com/quotes/about/education/) [Euler](https://sms4smile.com/quotes/about/euler/) [Giving](https://sms4smile.com/quotes/about/giving/) [Giving up](https://sms4smile.com/quotes/about/giving-up/) [Ideas](https://sms4smile.com/quotes/about/ideas/) [Learning](https://sms4smile.com/quotes/about/learning/) [Math](https://sms4smile.com/quotes/about/math/) [Mathematics](https://sms4smile.com/quotes/about/mathematics/) [Relevance](https://sms4smile.com/quotes/about/relevance/) [Understanding](https://sms4smile.com/quotes/about/understanding/) ## Related quotes > [After exponential quantities the circular functions, sine and cosine, should be considered because they arise when imaginary quantities are involved in the exponential.](https://sms4smile.com/quotes/leonhard-euler/after-exponential-quantities-the-circular-functions-sine-and-cosine/) [Leonhard Euler](https://sms4smile.com/quotes/author/leonhard-euler/) > [... I soon found an opportunity to be introduced to a famous professor Johann Bernoulli. ... True, he was very busy and so refused…](https://sms4smile.com/quotes/leonhard-euler/i-soon-found-an-opportunity-to-be-introduced-to-a-famous-professor/) [Leonhard Euler](https://sms4smile.com/quotes/author/leonhard-euler/) > [In the meantime, most noble Sir, you have assigned this question to the geometry of position, but I am ignorant as to what this…](https://sms4smile.com/quotes/leonhard-euler/in-the-meantime-most-noble-sir-you-have-assigned-this-question-to-the/) [Leonhard Euler](https://sms4smile.com/quotes/author/leonhard-euler/) > [For the sake of brevity, we will always represent this number 2.718281828459... by the letter e.](https://sms4smile.com/quotes/leonhard-euler/for-the-sake-of-brevity-we-will-always-represent-this-number-2/) [Leonhard Euler](https://sms4smile.com/quotes/author/leonhard-euler/) > [If a nonnegative quantity was so small that it is smaller than any given one, then it certainly could not be anything but zero.…](https://sms4smile.com/quotes/leonhard-euler/if-a-nonnegative-quantity-was-so-small-that-it-is-smaller-than-any/) [Leonhard Euler](https://sms4smile.com/quotes/author/leonhard-euler/) > [Since the fabric of the universe is most perfect and the work of a most wise Creator, nothing at all takes place in the…](https://sms4smile.com/quotes/leonhard-euler/since-the-fabric-of-the-universe-is-most-perfect-and-the-work-of-a/) [Leonhard Euler](https://sms4smile.com/quotes/author/leonhard-euler/) > [The kind of knowledge which is supported only by observations and is not yet proved must be carefully distinguished from the truth; it is…](https://sms4smile.com/quotes/leonhard-euler/the-kind-of-knowledge-which-is-supported-only-by-observations-and-is/) [Leonhard Euler](https://sms4smile.com/quotes/author/leonhard-euler/) > [Thus you see, most noble Sir, how this type of solution to the Königsberg bridge problem bears little relationship to mathematics, and I do…](https://sms4smile.com/quotes/leonhard-euler/thus-you-see-most-noble-sir-how-this-type-of-solution-to-the-k/) [Leonhard Euler](https://sms4smile.com/quotes/author/leonhard-euler/) > [A function of a variable quantity is an analytic expression composed in any way whatsoever of the variable quantity and numbers or constant quantities.](https://sms4smile.com/quotes/leonhard-euler/a-function-of-a-variable-quantity-is-an-analytic-expression-composed/) [Leonhard Euler](https://sms4smile.com/quotes/author/leonhard-euler/) > [Although to penetrate into the intimate mysteries of nature and thence to learn the true causes of phenomena is not allowed to us, nevertheless…](https://sms4smile.com/quotes/leonhard-euler/although-to-penetrate-into-the-intimate-mysteries-of-nature-and/) [Leonhard Euler](https://sms4smile.com/quotes/author/leonhard-euler/) > [The person who did most to give to analysis the generality and symmetry which are now its pride, was also the person who made…](https://sms4smile.com/quotes/william-whewell/the-person-who-did-most-to-give-to-analysis-the-generality-and/) [William Whewell](https://sms4smile.com/quotes/author/william-whewell/) > [“I have been able to solve a few problems of mathematical physics on which the greatest mathematicians since Euler have struggled in vain ...…](https://sms4smile.com/quotes/hermann-von-helmholtz/i-have-been-able-to-solve-a-few-problems-of-mathematical-physics-on/) [Hermann von Helmholtz](https://sms4smile.com/quotes/author/hermann-von-helmholtz/) > [Euler calculated without effort, just as men breathe, as eagles sustain themselves in the air.](https://sms4smile.com/quotes/francois-arago/euler-calculated-without-effort-just-as-men-breathe-as-eagles-sustain/) [Francois Arago](https://sms4smile.com/quotes/author/francois-arago/) > [“Along with working on the Basel problem, Euler realized that adding an infinite sequence of reciprocal powers for all whole numbers will give you…](https://sms4smile.com/quotes/matt-parker/along-with-working-on-the-basel-problem-euler-realized-that-adding-an/) [Matt Parker](https://sms4smile.com/quotes/author/matt-parker/) > [“Durante la invasión rusa en 1760, la casa de campo de Euler fue saqueada. El general ruso declaró que "no hacía la guerra a…](https://sms4smile.com/quotes/anonymous/durante-la-invasion-rusa-en-1760-la-casa-de-campo-de-euler-fue/) [Anonymous](https://sms4smile.com/quotes/author/anonymous/) ## More by Leonhard Euler > [Nothing takes place in the world whose meaning is not that of some maximum or minimum.](https://sms4smile.com/quotes/leonhard-euler/nothing-takes-place-in-the-world-whose-meaning-is-not-that-of-some/) [Leonhard Euler](https://sms4smile.com/quotes/author/leonhard-euler/) > [Although to penetrate into the intimate mysteries of nature and thence to learn the true causes of phenomena is not allowed to us, nevertheless…](https://sms4smile.com/quotes/leonhard-euler/although-to-penetrate-into-the-intimate-mysteries-of-nature-and/) [Leonhard Euler](https://sms4smile.com/quotes/author/leonhard-euler/) > [A function of a variable quantity is an analytic expression composed in any way whatsoever of the variable quantity and numbers or constant quantities.](https://sms4smile.com/quotes/leonhard-euler/a-function-of-a-variable-quantity-is-an-analytic-expression-composed/) [Leonhard Euler](https://sms4smile.com/quotes/author/leonhard-euler/) > [Thus you see, most noble Sir, how this type of solution to the Königsberg bridge problem bears little relationship to mathematics, and I do…](https://sms4smile.com/quotes/leonhard-euler/thus-you-see-most-noble-sir-how-this-type-of-solution-to-the-k/) [Leonhard Euler](https://sms4smile.com/quotes/author/leonhard-euler/) > [The kind of knowledge which is supported only by observations and is not yet proved must be carefully distinguished from the truth; it is…](https://sms4smile.com/quotes/leonhard-euler/the-kind-of-knowledge-which-is-supported-only-by-observations-and-is/) [Leonhard Euler](https://sms4smile.com/quotes/author/leonhard-euler/) > [Since the fabric of the universe is most perfect and the work of a most wise Creator, nothing at all takes place in the…](https://sms4smile.com/quotes/leonhard-euler/since-the-fabric-of-the-universe-is-most-perfect-and-the-work-of-a/) [Leonhard Euler](https://sms4smile.com/quotes/author/leonhard-euler/) > [If a nonnegative quantity was so small that it is smaller than any given one, then it certainly could not be anything but zero.…](https://sms4smile.com/quotes/leonhard-euler/if-a-nonnegative-quantity-was-so-small-that-it-is-smaller-than-any/) [Leonhard Euler](https://sms4smile.com/quotes/author/leonhard-euler/) > [Logic is the foundation of the certainty of all the knowledge we acquire.](https://sms4smile.com/quotes/leonhard-euler/logic-is-the-foundation-of-the-certainty-of-all-the-knowledge-we/) [Leonhard Euler](https://sms4smile.com/quotes/author/leonhard-euler/) > [For the sake of brevity, we will always represent this number 2.718281828459... by the letter e.](https://sms4smile.com/quotes/leonhard-euler/for-the-sake-of-brevity-we-will-always-represent-this-number-2/) [Leonhard Euler](https://sms4smile.com/quotes/author/leonhard-euler/) > [In the meantime, most noble Sir, you have assigned this question to the geometry of position, but I am ignorant as to what this…](https://sms4smile.com/quotes/leonhard-euler/in-the-meantime-most-noble-sir-you-have-assigned-this-question-to-the/) [Leonhard Euler](https://sms4smile.com/quotes/author/leonhard-euler/) > [For since the fabric of the universe is most perfect and the work of a most wise Creator, nothing at all takes place in…](https://sms4smile.com/quotes/leonhard-euler/for-since-the-fabric-of-the-universe-is-most-perfect-and-the-work-of/) [Leonhard Euler](https://sms4smile.com/quotes/author/leonhard-euler/) > [Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe…](https://sms4smile.com/quotes/leonhard-euler/mathematicians-have-tried-in-vain-to-this-day-to-discover-some-order/) [Leonhard Euler](https://sms4smile.com/quotes/author/leonhard-euler/) > [Now I will have less distraction.](https://sms4smile.com/quotes/leonhard-euler/now-i-will-have-less-distraction/) [Leonhard Euler](https://sms4smile.com/quotes/author/leonhard-euler/) > [ Transcendental \[numbers\], They transcend the power of algebraic methods. ](https://sms4smile.com/quotes/leonhard-euler/transcendental-numbers-they-transcend-the-power-of-algebraic-methods/) [Leonhard Euler](https://sms4smile.com/quotes/author/leonhard-euler/) > [To those who ask what the infinitely small quantity in mathematics is, we answer that it is actually zero. Hence there are not so…](https://sms4smile.com/quotes/leonhard-euler/to-those-who-ask-what-the-infinitely-small-quantity-in-mathematics-is/) [Leonhard Euler](https://sms4smile.com/quotes/author/leonhard-euler/) [Explore all 19 Leonhard Euler quotes ](https://sms4smile.com/quotes/author/leonhard-euler/) ## More Controversy quotes > [If Melanchthon were alive today, he might not weep because of controversies that surround the Lord's Supper, but he might well sorrow because of…](https://sms4smile.com/quotes/erwin-w-lutzer/if-melanchthon-were-alive-today-he-might-not-weep-because-of/) [Erwin W. Lutzer](https://sms4smile.com/quotes/author/erwin-w-lutzer/) > [ As we have seen after every other \[Donald\] Trump controversy, this one only increased their enthusiasm for him. His supporters thought the idea of… ](https://sms4smile.com/quotes/steve-inskeep/as-we-have-seen-after-every-other-donald-trump-controversy-this-one/) [Steve Inskeep](https://sms4smile.com/quotes/author/steve-inskeep/) > [My pride is the length of time I have worked in the industry without any major controversy.](https://sms4smile.com/quotes/sajid-nadiadwala/my-pride-is-the-length-of-time-i-have-worked-in-the-industry-without/) [Sajid Nadiadwala](https://sms4smile.com/quotes/author/sajid-nadiadwala/) > [Controversy equalizes fools and wise men in the same way - and the fools know it.](https://sms4smile.com/quotes/oliver-wendell-holmes-sr-2/controversy-equalizes-fools-and-wise-men-in-the-same-way-and-the/) [Oliver Wendell Holmes, Sr](https://sms4smile.com/quotes/author/oliver-wendell-holmes-sr-2/) > [Some people want to have controversy between the races.](https://sms4smile.com/quotes/dennis-rodman/some-people-want-to-have-controversy-between-the-races/) [Dennis Rodman](https://sms4smile.com/quotes/author/dennis-rodman/) > [Controversy for the sake of controversy is sin. Controversy for the sake of truth is a divine command.](https://sms4smile.com/quotes/walter-martin/controversy-for-the-sake-of-controversy-is-sin-controversy-for-the/) [Walter Martin](https://sms4smile.com/quotes/author/walter-martin/) > [Sometimes as a player you do not want to get into a controversy.](https://sms4smile.com/quotes/sourav-ganguly/sometimes-as-a-player-you-do-not-want-to-get-into-a-controversy/) [Sourav Ganguly](https://sms4smile.com/quotes/author/sourav-ganguly/) > [Science does not enter a chaotic society to put order into it anymore, to simplify its composition, and to put an end to controversies.…](https://sms4smile.com/quotes/bruno-latour/science-does-not-enter-a-chaotic-society-to-put-order-into-it-anymore/) [Bruno Latour](https://sms4smile.com/quotes/author/bruno-latour/) > [I am disappointed by this controversy surrounding A Million Little Pieces because I rely on the publishers to define the category that a book…](https://sms4smile.com/quotes/oprah-winfrey/i-am-disappointed-by-this-controversy-surrounding-a-million-little/) [Oprah Winfrey](https://sms4smile.com/quotes/author/oprah-winfrey/) > [My stress on the truth in sexual stereotypes and on the biologic basis of sex differences is sure to cause controversy.](https://sms4smile.com/quotes/camille-paglia/my-stress-on-the-truth-in-sexual-stereotypes-and-on-the-biologic/) [Camille Paglia](https://sms4smile.com/quotes/author/camille-paglia/) > [Controversy sells, politics sell, all that type of stuff sells.](https://sms4smile.com/quotes/jadakiss/controversy-sells-politics-sell-all-that-type-of-stuff-sells/) [Jadakiss](https://sms4smile.com/quotes/author/jadakiss/) > [Please don't think that I got married on TV for publicity. I already enjoyed a huge fan following, courtesy my character in Bidaai.' Instead,…](https://sms4smile.com/quotes/sara-khan/please-don-t-think-that-i-got-married-on-tv-for-publicity-i-already/) [Sara Khan](https://sms4smile.com/quotes/author/sara-khan/) > [There's a lot of controversy online, some people say i'm a genius and other say i'm hugely talented.](https://sms4smile.com/quotes/andy-kindler/there-s-a-lot-of-controversy-online-some-people-say-i-m-a-genius-and/) [Andy Kindler](https://sms4smile.com/quotes/author/andy-kindler/) > [Gossip is never fatal until it is denied. Gossip goes on about every human being alive and about all the dead that are alive…](https://sms4smile.com/quotes/booth-tarkington/gossip-is-never-fatal-until-it-is-denied-gossip-goes-on-about-every/) [Booth Tarkington](https://sms4smile.com/quotes/author/booth-tarkington/) > [I think controversy is an interesting thing in that it usually goes hand in hand with starting conversations that surround taboo or under-discussed topics…](https://sms4smile.com/quotes/alex-sharp/i-think-controversy-is-an-interesting-thing-in-that-it-usually-goes/) [Alex Sharp](https://sms4smile.com/quotes/author/alex-sharp/) [Browse all 604 Controversy quotes ](https://sms4smile.com/quotes/about/controversy/) ```json { "@context": "https://schema.org", "@graph": [ { "@type": "Quotation", "text": "Notable enough, however, are the controversies over the series 1 - 1 + 1 - 1 + 1 - ... whose sum was given by Leibniz as 1/2, although others disagree. ... 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