**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.

The inequality states that [1] if *g* is monotone, then |Σ_{x≤m} *g*(*x*) *f*(*x*)| is bounded above by:

max_{x≤m} |*g*(*x*)| max_{j≤m} |Σ_{x≤j} *f*(*x*)|.

To put this another way:

If, for all values of n,

A > u_{1} + u_{2} + … + u_{n} > B,

Where u_{i} are real-valued quantities, and if a_{1}, a_{2}, …, a_{n} are a series of positive quantities that constantly decrease as n increases, then:

a_{1}A > a_{1} u_{1} + a_{2} u_{2} + … a_{n} u_{n} > a_{1}B.

It can also be stated as [2]

Let z_{1}, …, z_{n} ∈ ℂ, and let S_{k} = z_{1} + … + z_{k}, for all k. Then, for each sequence of real numbers a_{1} ≥ a_{2} ≥ … ≥ a_{n} ≥ 0:

## Proof of Abel’s Inequality

We have a_{n} = s_{n} − s_{n} − 1, so [3]:

= s_{1}b_{1} + (s_{2} − s_{1})b_{2} + … + (s_{m} − s_{m} − 1)b_{m} = s_{1}(b_{1} − b_{2}) + s_{2}(b_{2} − b_{3}) + … + s/sub>m−1(b_{m−1} − b_{m}) + s_{m}b_{m}.

Every |s_{n}| ≤ M and b_{n} − b_{n + 1} is positive, so

|s_{n}(b_{n} − b_{n+1})| ≤ M(b_{n} − b_{n+1}).

By the triangle inequality&:

≤ |s_{1}(b_{1} − b_{2})| + … + |s_{m−1}(b_{m−1} − b_{m})| + |s_{m}b_{m}| ≤ M(b_{1} − b_{2}) + … + M(b_{m−1} − b_{m}) + M b_{m} = M b_{1}.

*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).

## References

[1] Wolf, J. Sets Whose Difference Set is Square-Free. Retrieved July 3, 2021 from: https://www.cs.umd.edu/~gasarch/TOPICS/vdw/wolfsq.pdf

[2] 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

[3] Azoff, E. (2010). Sequences and Series. Retrieved July 3, 2021 from: http://alpha.math.uga.edu/~azoff/courses/3100sp10.pdf

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