What is the value of the sum ? Furdui poses this as Question 1.28 in his book Limits, Series, and Fractional Part Integrals [1, p. 5], where he calls the sum a “double Euler-Mascheroni sequence.” His solution converts the double sum into an expression containing an integral and an infinite series, and then he evaluates those. In this post I’ll give a different derivation of Furdui’s expression for the value of the double sum – one that uses only standard summation manipulations.
Let denote the nth harmonic number: . The following lemma is proved in Concrete Mathematics [2, pp. 40-41] and is a straightforward application of summation by parts, but I’ll give a proof here anyway.
Proof: Swapping the order of summation, we have
With the lemma in place, we’re now ready to derive an expression for this double Euler-Mascheroni sequence.
Proof: We have .
Applying the lemma, we obtain
- Ovidiu Furdui, Limits, Series, and Fractional Part Integrals, Springer, 2013.
- Ronald L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete Mathematics (2nd ed.), Addison-Wesley, 1994.