Abel’s Inequality, named after 19th Century Norwegian mathematician Niels Abel, is an estimate for the sum of products of two numbers. It’s uses include investigating convergence of sums and showing that certain partial sums form a Cauchy sequence.
To put this another way:
If, for all values of n,
A > u1 + u2 + … + un > B,
Where ui are real-valued quantities, and if a1, a2, …, an are a series of positive quantities that constantly decrease as n increases, then:
a1A > a1 u1 + a2 u2 + … an un > a1B.
It can also be stated as 
Let z1, …, zn ∈ ℂ, and let Sk = z1 + … + zk, for all k. Then, for each sequence of real numbers a1 ≥ a2 ≥ … ≥ an ≥ 0:
Proof of Abel’s Inequality
We have an = sn − sn − 1, so :
= s1b1 + (s2 − s1)b2 + … + (sm − sm − 1)bm = s1(b1 − b2) + s2(b2 − b3) + … + s/sub>m−1(bm−1 − bm) + smbm.
Every |sn| ≤ M and bn − bn + 1 is positive, so
|sn(bn − bn+1)| ≤ M(bn − bn+1).
By the triangle inequality&:
≤ |s1(b1 − b2)| + … + |sm−1(bm−1 − bm)| + |smbm| ≤ M(b1 − b2) + … + M(bm−1 − bm) + M bm = M b1.
*The Triangle Inequality d(x, y) + d(y, z) ≥ g(x, z) states that the third side of a triangle is always less than the sum of its two sides (or equal, in the case that all points are on a straight line).
 Wolf, J. Sets Whose Difference Set is Square-Free. Retrieved July 3, 2021 from: https://www.cs.umd.edu/~gasarch/TOPICS/vdw/wolfsq.pdf
 Ferber, A. (2020). Arbitrary Topics. Retrieved July 3, 2021 from: https://cpb-us-e2.wpmucdn.com/faculty.sites.uci.edu/dist/f/842/files/2020/06/Inequalities.pdf
 Azoff, E. (2010). Sequences and Series. Retrieved July 3, 2021 from: http://alpha.math.uga.edu/~azoff/courses/3100sp10.pdf
Stephanie Glen. "Abel’s Inequality: Definition & Proof" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/abels-inequality-definition-proof/
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