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In this section of Resonance, we invite readers to pose questions likely to be raised in a classroom situation. We may suggest strategies for dealing with them, or invite responses, or both. “Classroom” is equally a forum for raising broader issues and sharing personal experiences and viewpoints on matters related to teaching and learning science.
Euler’s Summation Method∗
Kapil Hari Paranjape Indian Institute of Science Education and Research Mohali Sector 81, SAS Nagar Manauli PO
What is the meaning of an infinite sum? This question has fascinated mathematicians for a long time; from Zeno’s paradox and the series of Madhava and Leibnitz to more contemporary times. Euler, Fourier and others played insouciantly with infinite series until Abel, Cauchy and Weierstrass gave us a safe and dependable way to “do the right thing” with infinite series. Like other such instances in Mathematics, this did not shut the door on the older playground. Rather it provided a framework to play in it with greater clarity.
Punjab 140 306, India. Email:
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The author learned a lot about this the topic through the book on Divergent Series by G. H. Hardy, while teaching a course on “Computational Methods” at IISER Mohali. Through “abuse of notation” we often write infinite sums in the form a1 + a2 + · · ·. This is called an abuse of notation as it is actually not immediately obvious what this means! In this note we will discuss some ways to define and find the value of such sums.
Keywords Infinite sums, convergence, accuracy.
In school, we begin by learning how to add two integers and then ∗
Vol.25, No.7, DOI: https://doi.org/10.1007/s12045-020-1017-8
RESONANCE | July 2020
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CLASSROOM
two fractions. It is trivial to check (but not obvious) that this allows us to assign a unique value to expressions like 1+1/2+1/3. Should we group the 1 and the 1/2 to get 3/2 and then add the 1/3 to get 11/6, or should we group 1/2 and 1/3 to get 5/6 and then add this to 1 to get 11/6? As the example shows (and one can prove!), it does not matter how we group the numbers. The equation: (a1 + a2 ) + a3 = a1 + (a2 + a3 ) is the Associative law for addition of numbers and we use it all the time to avoid writing a large number of brackets! Note that this does not work in simple day-to-day tasks. If you compare warm coffee decoction + hot milk + sugar with
(warm coffee decoction + hot milk) + sugar, you will realise that it is much harder to dissolve the sugar in the second case since the mixture of coffee and milk is no longer hot enough! One gets a unique value for a1 + a2 + · · · + an regardless of grouping. The situation for infinite sums is completely different.
This is not a problem with numbers so one gets a unique value for a1 + a2 + · · · + an regardless of how we group the numbers to perform the addition pairwise.
Infinite Sums? The situation changes completely when we try to make sense of an infinite sum. The first thing to recognise is that in practice we can only add finitely many terms! So, the notion of
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