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 *n*th 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.

**Lemma**: .

*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.

**Theorem**: .

*Proof*: We have .

Applying the lemma, we obtain

**References**

- 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.

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