Fast convergence of generalized DeTemple sequences and the relation to the Riemann zeta function

There are many famous unsolved problems associated with the properties of this constant; for example it is not known yet if the Euler constant is irrational. The Euler constant is an unusual constant and it seems unrelated to other known constants. However, there are many areas in which Euler’s constant appears, for example, probability theory, random matrix theory and Riemann hypothesis, etc. Recently, Agarwal et al. [5], Wang et al. [6–9] and Yang et al. [10] showed that the Euler-Mascheroni constant has important applications in the field of special functions. In a sense, its appearance in different mathematical areas can be regarded as possible connections between these subjects.

If \(\lim_{n\to \infty }x_{n}=x\), with \(x\in \mathbb {R}\), and \(\lim_{n\to \infty }{b_{n}\over n}=0\), then the sequence \((\omega _{n})_{n\geq1}\) is convergent to \(\gamma +x\).

To calculate the rate of convergence, we will utilize the Stolz lemma in the case of \({0\over 0}\).

We shall write the difference \(\omega _{n}-\omega _{n-1}\) as a power series of \(n^{-1}\).

The above relational expression (1.1) is useful, it will be used to establish our main results in subsequent sections.

Let us consider certain special cases of \(x_{n}-x_{n-1}\) and \(b_{n}-b_{n-1}\), which will enable us to establish the generalizations of some sequences from DeTemple [1] and Mortici [11].

Case 1. Suppose \(x_{n}-x_{n-1}=0\), for all \(n\geq2\) and \(b_{n}-b_{n-1}=0\), for all \(n\geq2\).

In this case the sequences \((x_{n})_{n\geq1}\) and \((b_{n})_{n\geq1}\) are constant; let \(x_{n}=c\), for all \(n\geq1\) and \(b_{n}=b\), for all \(n\geq1\).

In (1.1), the coefficient of \({1\over 2n^{2}}\) becomes \(b_{n}^{2}-(b_{n-1}-1)^{2}=b^{2}-(b-1)^{2}=2b-1\).

Subcase 1.2. If \(b\neq{1\over 2}\), then the term in \({1\over n^{2}}\) is present, therefore the speed of convergence is \(n^{-1}\).

For \(b=0\), we obtain the sequence defined in Theorem 3.1 of [11].

The speed of convergence is \(n^{-3}\). Note that the logarithm \(\log(n+b)\) has definition for \(b\geq0\). This leads to the following assertion.

For \(b=0\), we obtain the sequence defined in Theorem 3.2 of [11].

In this section we shall establish two generalized DeTemple sequences in terms of the condition \(b_{n}+b_{n-1}-1=0\).

Note that the logarithm function in the above expression has definition for \(b_{1}\in(0,1)\). Hence, we have the following result.

It is clear that for \(b_{1}={1\over 2}\) we obtain DeTemple sequence.

As above, we consider the sequence \(b_{n}={(-1)^{n-1}(2b_{1}-1)+1\over 2}\), \(b_{1}\in(0,1)\).

Putting \(a_{k}={(-1)^{k-1}(2b_{1}-1)+1\over 2k}\), then \(\lim_{k\to \infty }a_{k}=0\).

Consequently, we get the following result.

It is easy to observe that \(a_{2k}={1-b_{1}\over 2k}\) and \(a_{2k+1}={b_{1}\over 2k+1}\).

In order to find the limit of the sequence \((\omega _{n})_{n\geq1}\), we note that it is related to the harmonic sequence \(\zeta_{n}(s)=\sum_{k=1}^{n}{1\over k^{s}}\).

Thus the sequence \((\zeta_{n}(s))_{n\geq1}\) converges with speed of convergence \(n^{1-s}\).

We have showed in Case 1 that \(\lim_{n\to \infty }\sum_{k=1}^{n}{2a_{k}}-\log (n+na_{n})=\gamma +(2b_{1}-1)\log2\).

Therefore, we obtain the following theorem.

Thus, we obtain the following assertion.

Particular cases. For \(p=0\), the sequence \((\omega _{n})_{n\geq1}\) is a DeTemple sequence.

For \(p=1\), the sequence \((\omega _{n})_{n\geq1}\) is the sequence defined in Theorem 4.1 of [12].

For \(p=2\), the sequence \((\omega _{n})_{n\geq1}\) is the sequence defined in Theorem 4.2 of [12].

We remark that the expressions from Theorem 3.3 and Corollary 3.1 concern the expansion of Euler’s constant in terms of the Riemann zeta function evaluated at positive odd integers. Therefore the constant γ can be approximated with a very high speed of convergence by the above expansions. For more details of the series representations of the Euler constant, we mention the work of Alzer, Karayannakis and Srivastava [12], Alzer and Koumandos [13] and Sofo [14].

Our method for accelerating the convergence of DeTemple type sequences is more general than the method introduced in [11] and [15] by Mortici. We support this statement by the large degree of freedom in choosing the sequences \((x_{n})_{n\geq1}\), and \((b_{n})_{n\geq1}\). In this way, we generalized sequences from [11] and introduced two new sequences which generalized DeTemple sequence \((R_{n})_{n\geq1}\) having faster convergence rate.