Alternating harmonic series containing floor function
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Let $$Sleft(aright)=sum_n=0^infty fracleft(-1right)^left[naright]left[naright]+1$$
where $agt 0$, and $left[cdotright]$ denotes the floor function.
Consider $ain mathbbQ$, we can conclude that not all $a$ can make $Sleft(aright)$ converge.
In particular, we have $Sleft(frac1kright)=kln 2$.
Let $$A=leftamid Sleft(aright):convergesright$$
My questions are:
•Does $A$ contain any irrational numbers?
•Is $A$ an uncountable set? A full-measure set?
•If so, I would guess the following limit $$lim_ain A, arightarrow 0^+a, Sleft(aright)$$ exists and gets $ln 2$. Am I right?
real-analysis sequences-and-series analysis floor-function
asked Aug 9 at 16:58
Antimonius
3017
add a comment |Â
up vote
3
down vote
favorite
Let $$Sleft(aright)=sum_n=0^infty fracleft(-1right)^left[naright]left[naright]+1$$
where $agt 0$, and $left[cdotright]$ denotes the floor function.
Consider $ain mathbbQ$, we can conclude that not all $a$ can make $Sleft(aright)$ converge.
In particular, we have $Sleft(frac1kright)=kln 2$.
Let $$A=leftamid Sleft(aright):convergesright$$
My questions are:
•Does $A$ contain any irrational numbers?
•Is $A$ an uncountable set? A full-measure set?
•If so, I would guess the following limit $$lim_ain A, arightarrow 0^+a, Sleft(aright)$$ exists and gets $ln 2$. Am I right?
real-analysis sequences-and-series analysis floor-function
asked Aug 9 at 16:58
Antimonius
3017
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
Let $$Sleft(aright)=sum_n=0^infty fracleft(-1right)^left[naright]left[naright]+1$$
where $agt 0$, and $left[cdotright]$ denotes the floor function.
Consider $ain mathbbQ$, we can conclude that not all $a$ can make $Sleft(aright)$ converge.
In particular, we have $Sleft(frac1kright)=kln 2$.
Let $$A=leftamid Sleft(aright):convergesright$$
My questions are:
•Does $A$ contain any irrational numbers?
•Is $A$ an uncountable set? A full-measure set?
•If so, I would guess the following limit $$lim_ain A, arightarrow 0^+a, Sleft(aright)$$ exists and gets $ln 2$. Am I right?
real-analysis sequences-and-series analysis floor-function
asked Aug 9 at 16:58
Antimonius
3017
Let $$Sleft(aright)=sum_n=0^infty fracleft(-1right)^left[naright]left[naright]+1$$
where $agt 0$, and $left[cdotright]$ denotes the floor function.
Consider $ain mathbbQ$, we can conclude that not all $a$ can make $Sleft(aright)$ converge.
In particular, we have $Sleft(frac1kright)=kln 2$.
Let $$A=leftamid Sleft(aright):convergesright$$
My questions are:
•Does $A$ contain any irrational numbers?
•Is $A$ an uncountable set? A full-measure set?
•If so, I would guess the following limit $$lim_ain A, arightarrow 0^+a, Sleft(aright)$$ exists and gets $ln 2$. Am I right?
real-analysis sequences-and-series analysis floor-function
asked Aug 9 at 16:58
Antimonius
3017
edited Aug 10 at 14:06
asked Aug 9 at 16:58
Antimonius
3017
asked Aug 9 at 16:58
Antimonius
3017
asked Aug 9 at 16:58
Antimonius
3017
3017
add a comment |Â
add a comment |Â
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