In this section we shall establish two generalized DeTemple sequences in terms of the condition \(b_{n}+b_{n-1}-1=0\).
Case 1. Suppose
$$ x_{n}-x_{n-1}-{b_{n}-b_{n-1}\over n}+ {b_{n}^{2}-(b_{n-1}-1)^{2}\over 2n^{2}}=0 \quad (n\geq2). $$
(3.1)
We would like to find the sequence \((b_{n})_{n\geq1}\) such that \(b_{n}+b_{n-1}-1=0\) (this vanishes; the nominator \(b_{n}^{2}-(b_{n-1}-1)^{2}=0\)). It follows that \(b_{n}=-b_{n-1}+1\). Therefore \(b_{n}={(-1)^{n-1}\cdot(2b_{1}-1)+1\over 2}\), and this satisfies the initial condition \(\lim_{n\to \infty }{b_{n}\over n}=0\). On the other hand, one has \(b_{n}-b_{n-1}=(-1)^{n-1}(2b_{1}-1)\).
Equation (3.1) becomes
$$x_{n}-x_{n-1}={(-1)^{n-1}\over n} (2b_{1}-1). $$
Taking the sum from 2 to n and fixing \(x_{1}=2b_{1}-1\), we obtain
$$x_{n}=(2b_{1}-1) \biggl({1\over 1}- {1\over 2}+{1\over 3}-{1\over 4}+\cdots+ {(-1)^{n-1}\over n} \biggr). $$
We remark that \((x_{n})_{n\geq1}\) is the Leibniz’s sequence multiplied by the constant \((2b_{1}-1)\). This sequence is convergent, having the limit
$$\lim_{n\to \infty } x_{n}=(2b_{1}-1) \log2. $$
Therefore, we obtain the sequence
$$\begin{aligned} \omega _{n} =&1+{1\over 2}+\cdots+{1\over n}+(2b_{1}-1) \biggl(1-{1\over 2}+{1\over 3}-{1\over 4}+ \cdots+{(-1)^{n-1}\over n} \biggr) \\ &{}-\log \biggl[n+{(-1)^{n-1}\cdot(2b_{1}-1)+1\over 2} \biggr], \end{aligned}$$
or, in more suitable form,
$$ \omega _{n}=\sum_{k=1}^{n} {1+(2b_{1}-1)(-1)^{k-1}\over k}-\log \biggl[n+{(-1)^{n-1}\cdot(2b_{1}-1)+1\over 2} \biggr], $$
(3.2)
with speed of convergence \(n^{-2}\) and limit \(\gamma +(2b_{1}-1)\log2\).
Note that the logarithm function in the above expression has definition for \(b_{1}\in(0,1)\). Hence, we have the following result.
Theorem 3.1
If we denote by
\((w_{n})_{n\geq1}\)
the sequence (3.2), then
\((w_{n})_{n\geq1}\)
converges quadratically to its limit
\(\gamma +(2b_{1}-1)\log2\).
It is clear that for \(b_{1}={1\over 2}\) we obtain DeTemple sequence.
Case 2. Note that
$$\begin{aligned} \omega _{n}-\omega _{n-1} =&x_{n}-x_{n-1}- {b_{n}-b_{n-1}\over n}-{b_{n}^{3}-(b_{n-1}-1)^{3}\over 3n^{3}} \\ &{}- \cdots-{b_{n}^{2p+1}-(b_{n-1}-1)^{2p+1}\over (2p+1)n^{2p+1}}+O \biggl(\frac{1}{n^{2p+3}} \biggr). \end{aligned}$$
As above, we consider the sequence \(b_{n}={(-1)^{n-1}(2b_{1}-1)+1\over 2}\), \(b_{1}\in(0,1)\).
We have seen that it satisfies the equality \(b_{n}+b_{n-1}-1=0\). Then \(b_{n}^{2}-(b_{n-1}-1)^{2}=0\) and, in general,
$$b_{n}^{2k}-(b_{n-1}-1)^{2k}=0, \quad k=1,2,\ldots. $$
Suppose
$$x_{n}-x_{n-1}={b_{n}-b_{n-1}\over n}+{b_{n}^{3}-(b_{n-1}-1)^{3}\over 3n^{3}}+ \cdots +{b_{n}^{2p+1}-(b_{n-1}-1)^{2p+1}\over (2p+1)n^{2p+1}}. $$
From \(b_{n}+b_{n-1}-1=0\), one has \(b_{n-1}-1=-b_{n}\). Further, it follows that
$$b_{n}^{2k+1}-({b_{n-1}-1})^{2k+1}=b_{n}^{2k+1}-(-b_{n})^{2k+1} =2b_{n}^{2k+1}. $$
On the other hand,
$$\begin{aligned} {b_{n}-b_{n-1}\over n} =& {b_{1}(-1)^{n-1}+{1-(-1)^{n-1}\over 2}-b_{1}(-1)^{n-2}-{1-(-1)^{n-2}\over 2}\over n} \\ =&{(-1)^{n-1}(2b_{1}-1)\over n}. \end{aligned}$$
Hence, we deduce that
$$x_{n}-x_{n-1}={(-1)^{n-1}(2b_{1}-1)\over n}+{2b_{n}^{3}\over 3n^{3}}+ {2b_{n}^{5}\over 5n^{5}}+\cdots+{2b_{n}^{2p+1}\over (2p+1)n^{2p+1}}. $$
Taking the sum for k from 2 to n, we have
$$x_{n}-x_{1}=\sum_{k=2}^{n} \biggl[{(-1)^{k-1}(2b_{1}-1)\over k}+{2b_{k}^{3}\over 3k^{3}}+\cdots+ {2b_{k}^{2p+1}\over (2p+1)k^{2p+1}} \biggr]. $$
If we fix
$$x_{1}={2b_{1}-1}+{2b_{1}^{3}\over 3}+ {2b_{1}^{5}\over 5}+\cdots+{2b_{1}^{2p+1}\over 2p+1}, $$
then we obtain
$$x_{n}=\sum_{k=1}^{n} \biggl[ {(-1)^{k-1}(2b_{1}-1)\over k}+{2b_{k}^{3}\over 3k^{3}}+\cdots+{2b_{k}^{2p+1}\over (2p+1)k^{2p+1}} \biggr]. $$
Hence,
$$\begin{aligned} \omega _{n} =&1+{1\over 2}+\cdots+{1\over n}+ \sum_{k=1}^{n} \biggl[{(-1)^{k-1}(2b_{1}-1)\over k}+ {2b_{k}^{3}\over 3k^{3}}+\cdots +{2b_{k}^{2p+1}\over (2p+1)k^{2p+1}} \biggr] \\ &{}-\log \biggl[n+{(-1)^{n-1}(2b_{1}-1)+1\over 2} \biggr] \\ =&\sum_{k=1}^{n}{1\over k}+ \sum_{k=1}^{n} \biggl[{(-1)^{k-1}(2b_{1}-1)\over k}+ {2b_{k}^{3}\over 3k^{3}}+\cdots+{2b_{k}^{2p+1}\over (2p+1)k^{2p+1}} \biggr] \\ &{}-\log \biggl[n+{(-1)^{n-1}(2b_{1}-1)+1\over 2} \biggr] \\ =&\sum_{k=1}^{n}\biggl[2\cdot {(-1)^{k-1}(2b_{1}-1)+1\over 2k}+{2\over 3} \biggl({(-1)^{k-1}(2b_{1}-1)+1\over 2k} \biggr)^{3} \\ &{}+\cdots+{2\over 2p+1} \biggl({(-1)^{k-1}(2b_{1}-1)+1\over 2k} \biggr)^{2p+1}\biggr] \\ &{}-\log \biggl[n+{(-1)^{n-1}(2b_{1}-1)+1\over 2} \biggr]. \end{aligned}$$
Putting \(a_{k}={(-1)^{k-1}(2b_{1}-1)+1\over 2k}\), then \(\lim_{k\to \infty }a_{k}=0\).
Further,
$$ \omega _{n} = 2\sum_{k=1}^{n} \biggl( {a_{k}\over 1}+{a_{k}^{3}\over 3}+\cdots+{a_{k}^{2p+1}\over 2p+1} \biggr)-\log (n+n a_{n} ). $$
Consequently, we get the following result.
Theorem 3.2
The sequence
$$ \omega _{n} = 2\sum_{k=1}^{n} \biggl( {a_{k}\over 1}+{a_{k}^{3}\over 3}+\cdots+{a_{k}^{2p+1}\over 2p+1} \biggr)-\log (n+n a_{n} ) $$
has speed of convergence
\(n^{-2p-2}\), where
\(a_{k}={(-1)^{k-1}(2b_{1}-1)+1\over 2k}\), \(b_{1}\in(0,1)\).
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 \(\zeta_{n}(s)=\sum_{k=1}^{n}{1\over k^{s}}\).
For \(s>1\), the limit of this sequence defined the celebrated Riemann zeta function \(\zeta(s)\), which is very important in mathematics. The speed of convergence of this sequence to its limit is described by the double inequality
$${1\over (s-1)(n+1)^{s-1}}< \zeta(s)-\zeta_{n}(s)< {1\over (s-1)n^{s-1}}. $$
Thus the sequence \((\zeta_{n}(s))_{n\geq1}\) converges with speed of convergence \(n^{1-s}\).
A direct calculation gives
$$\begin{aligned} \sum_{k=1}^{\infty}{2a_{k}^{2s+1}\over 2s+1} =& {2\over 2s+1} \bigl(a_{1}^{2s+1}+a_{2}^{2s+1}+ \cdots+a_{n}^{2s+1}+\cdots \bigr) \\ =&{2\over 2s+1} \biggl[ \biggl({b_{1}\over 1} \biggr)^{2s+1}+ \biggl({1-b_{1}\over 2} \biggr)^{2s+1}+ \biggl({b_{1}\over 3} \biggr)^{2s+1}+ \biggl( {1-b_{1}\over 4} \biggr)^{2s+1}+\cdots \biggr] \\ =&{2\over 2s+1}\biggl[b_{1}^{2s+1}\cdot \biggl( {1\over 1^{2s+1}}+{1\over 3^{2s+1}}+{1\over 5^{2s+1}}+\cdots \biggr) \\ &{}+(1-b_{1})^{2s+1}\cdot \biggl({1\over 2^{2s+1}}+ {1\over 4^{2s+1}}+{1\over 6^{2s+1}}+\cdots \biggr)\biggr] \\ =&{2\over 2s+1}\biggl[b_{1}^{2s+1} \biggl( {1\over 1^{2s+1}}+{1\over 2^{2s+1}}+{1\over 3^{2s+1}}+ {1\over 4^{2s+1}}+{1\over 5^{2s+1}}+\cdots \biggr) \\ &{}-b_{1}^{2s+1} \biggl({1\over 2^{2s+1}}+ {1\over 4^{2s+1}}+{1\over 6^{2s+1}}+\cdots \biggr) \\ &{}+(1-b_{1})^{2s+1}\cdot{1\over 2^{2s+1}} \biggl( {1\over 1^{2s+1}}+{1\over 2^{2s+1}}+{1\over 3^{2s+1}}+\cdots \biggr)\biggr]. \end{aligned}$$
Hence,
$$\begin{aligned} \sum_{k=1}^{\infty}{2a_{k}^{2s+1}\over 2s+1} =& {2\over 2s+1}\biggl[b_{1}^{2s+1}\cdot \zeta(2s+1)-b_{1}^{2s+1}\cdot{1\over 2^{2s+1}} \zeta(2s+1) \\ &{}+(1-b_{1})^{2s+1}\cdot{1\over 2^{2s+1}}\zeta(2s+1) \biggr] \\ =&{2\over 2s+1}\cdot {b_{1}^{2s+1}(2^{2s+1}-1)+(1-b_{1})^{2s+1}\over 2^{2s+1}}\cdot\zeta (2s+1) \\ =&{b_{1}^{2s+1}(2^{2s+1}-1)+(1-b_{1})^{2s+1}\over (2s+1)\cdot2^{2s}}\cdot\zeta(2s+1), \end{aligned}$$
where \(s\in\{1,2,\ldots,p\}\).
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.
Theorem 3.3
The sequence
\((\omega_{n})_{n\geq1}\)
defined by Theorem
3.2
converges to
$$\gamma +(2b_{1}-1)\log2+\sum_{s=1}^{p} {b_{1}^{2s+1}(2^{2s+1}-1)+(1-b_{1})^{2s+1}\over 2^{2s}(2s+1)}\zeta(2s+1) $$
with speed of convergence
\(n^{-2p-2}\).
If we take \(b_{1}={1\over 2}\) in Theorem 3.3, then \(a_{k}={1\over 2k}\),
$$\begin{aligned} \omega _{n} =&\sum_{k=1}^{n}2 \biggl[ {1\over 2k}+{1\over 3\cdot2^{3}k^{3}}+\cdots+{1\over (2p+1)2^{2p+1}k^{2p+1}} \biggr] \\ &{}-\log \biggl(n+{1\over 2} \biggr) \\ =&R_{n}+\sum_{k=1}^{n} \biggl[ {1\over 3\cdot2^{2} k^{3}}+\cdots+{1\over (2p+1) 2^{2p} k^{2p+1}} \biggr], \end{aligned}$$
(3.3)
where
$$R_{n}=1+{1\over 2}+\cdots+{1\over n}-\log \biggl(n+{1\over 2} \biggr). $$
Thus, we obtain the following assertion.
Corollary 3.1
The sequence
\((\omega _{n})_{n\geq1}\)
defined by (3.3) converges to
$$\gamma +\sum_{s=1}^{p}{1\over (2s+1)2^{2s}} \zeta(2s+1) $$
with speed of convergence
\(n^{-2p-2}\).
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].