Some remarks on faster convergent infinite series (Q2905560)

Let \((s_n)\) and \((s_n^*)\) be two sequences having limits \(s\) and \(s^*\), respectively. Then the sequence \((s_n^*)\) converges faster to its limit \(s^*\) than the sequence \((s_n)\) to its limit \(s\) if \(\lim _{n\rightarrow \infty }\frac {s_n^*-s^*}{s_n-s}=0\), with \(s_n-s \not =0\) for all \(n\in \mathbb N\).NEWLINENEWLINENEWLINELet \(\sum _{n=1}^\infty a_n\) and \(\sum _{n=1}^\infty b_n\) be infinite series with sequences of partial sums \(\bigl(s_n(a)\bigr)\) and \(\bigl (s_n(b)\bigr)\), respectively. Then \(\sum _{n=1}^\infty a_n\) converges faster than \(\sum _{n=1}^\infty b_n\) if \(\bigl(s_n(a)\bigr)\) converges faster than \(\left (s_n(b)\right)\).NEWLINENEWLINENEWLINE The authors have established the following results.NEWLINENEWLINE NEWLINEResult 1. Let \(\sum _{n=1}^\infty a_n\) be faster convergent series than (in short fcst) \(\sum _{n=1}^\infty b_n\) such that \(\lim _{n\rightarrow \infty }\frac {s_n(a)-s_{n-1}(a)}{s_n(b)-s_{n-1}(b)}=0\) and let \(\sum _{n=1}^\infty c_n\) be fcst \(\sum _{n=1}^\infty b_n\) such that \(\lim _{n\rightarrow \infty }\frac {s_n(c)-s_{n-1}(c)}{s_n(b) - s_{n-1}(b)} \not = 0\) or \(\lim _{n\rightarrow \infty}\frac {s_n(c)-s_{n-1}(c)}{s_n(b)-s_{n-1}(b)}\) does not exist and \(\liminf _{n\rightarrow \infty}| \frac {s_n(c)-s_{n-1}(c)}{s_n(b)-s_{n-1}(b)}| > 0\).NEWLINENEWLINENEWLINEIf \(\limsup _{n\rightarrow \infty}| \frac {s(a)-s_{n-1}(a)}{s_n(a)-s_{n-1}(a)}| < \infty\) and \(\liminf _{n\rightarrow \infty}| \frac {s(c)-s_{n-1}(c)}{s_n(c)-s_{n-1}(c)}| >0\), then \(\sum _{n=1}^\infty a_n\) is fcst \(\sum _{n=1}^\infty c_n\).NEWLINENEWLINENEWLINEResult 2. Let \(\sum _{n=1}^\infty b_n\) be a convergent series such that \(s(b)-s_{n-1}(b)\not =0\), for all \(n\in \mathbb N\).NEWLINENEWLINE(a) Then there exists \(\sum _{n=1}^\infty a_n\) fcst \(\sum _{n=1}^\infty b_n\) such that NEWLINENEWLINENEWLINE \(\lim _{n\rightarrow \infty}\frac {s_n(a)-s_{n-1}(a)}{s_n(b)-s_{n-1}(b)}=0\) and \(\limsup _{n\rightarrow \infty}| \frac {s(a)-s_{n-1}(a)}{s_n(a)-s_{n-1}(a)}| < \infty\).NEWLINENEWLINENEWLINE(b) The following are equivalent: NEWLINENEWLINENEWLINE{\parindent=1,5cm\begin{itemize}\item[(i)] there exists \(\sum _{n=1}^\infty a_n\) fcst \(\sum _{n=1}^\infty b_n\) such that \(\lim _{n\rightarrow \infty}\frac {s_n(a)-s_{n-1}(a)}{s_n(b)-s_{n-1}(b)}\) does not exist, NEWLINE\item[(ii)] \({\liminf _{n\rightarrow \infty }}| \frac {s_n(b)-s_{n-1}(b)}{s(b)-s_{n-1}(b)}| = 0\).NEWLINENEWLINE\end{itemize}} (c) Let \(\liminf _{n\rightarrow \infty}| \frac {s_n(b)-s_{n-1}(b)}{s(b)-s_{n-1}(b)}| > 0\) and \(\sum _{n=1}^\infty a_n\) be fcst \(\sum _{n=1}^\infty b_n\).NEWLINENEWLINEThen \(\lim _{n\rightarrow \infty }\frac {s_n(a)-s_{n-1}(a)}{s_n(b)-s_{n-1}(b)}=0\).