A New Infinite Series, Convergent and Irrational

Infinite series are a log of func. All kidding aside, you may have heard of the sum of reciprocal squares.

\sum\limits_{n=1}^{\infty}\frac{1}{{n}^2}=\frac{1}{{1}^2}+\frac{1}{{2}^2}+\frac{1}{{3}^2}+\frac{1}{{4}^2}+\frac{1}{{5}^2}+\frac{1}{{6}^2}+\frac{1}{{7}^2}+\cdots

The sum of this slowly convergent series is approximately equal to 1.644934. Is there anything special about this number? Italian mathematician Pietro Mengoli (1626-1686) first posed the question in 1644 (published 1650), what is the exact sum of this infinite series? This problem was not solved until 90 years later by Swiss mathematician Leonhard Euler (1707-1783) in 1734 (published 1735). Euler proved that the exact sum is

\sum\limits_{n=1}^{\infty}\frac{1}{{n}^2}=\frac{{\pi}^2}{6}

There’s that number pi, the most famous of the irrational numbers, showing up once again in mathematics, ostensibly having nothing at all to do with circles. What’s an irrational number? It is any real number that cannot be expressed as a ratio of two integers. The decimal digits of an irrational number neither terminate nor end in a repeating sequence (e.g. 1/3 = 0.3333… or 9/11 = 0.81818181…).

Determining the exact value of the sum of reciprocal squares infinite series is known as the Basel Problem, named after the hometown of Euler (who solved it) and the Bernoulli family of mathematicians (who were not able to solve it).

My admired colleague in England, Abdul Ahad, has come up with a variant of the sum of reciprocal squares where every third term starting with n = 4 is subtracted rather than added.

\frac{{\pi}^2}{6} -2\sum\limits_{n=1}^{\infty}\frac{1}{{\left (3n+1  \right )}^2}=\frac{1}{{1}^2}+\frac{1}{{2}^2}+\frac{1}{{3}^2}-\frac{1}{{4}^2}+\frac{1}{{5}^2}+\frac{1}{{6}^2}-\frac{1}{{7}^2}+\cdots

Ahad has shown that this new series is convergent and sums to an irrational number, approximately equal to 1.40146804. The infinite sum portion of the above expression is approximately equal to 0.12173301 and is also an irrational number. Multiplying by 2 gives us an irrational number, and subtracting from π2/6, which is itself irrational, results in our final result being irrational.

Interestingly, almost all real numbers are irrational, strange as they are.

References

Ahad, Abdul. “An interesting series.” M500 Magazine 278, 8-9 (2017). http://m500.org.uk/wp-content/uploads/2018/11/M278WEB.pdf.

Ahad, Abdul. “A New Infinite Series with Proof of Convergence and Irrational Sum.” Res Rev J Statistics Math Sci, Volume 4, Issue 1 (2018). http://www.rroij.com/open-access/a-new-infinite-series-with-proof-of-convergence-and-irrational-sum.pdf.