User:PAR/Test2

Assuming:

• a_n > 0 for all n
• D_n > 0 for all n

Kummer's test defines:

${\displaystyle \rho _{n}=\left(D_{n}-D_{n+1}{\frac {a_{n+1}}{a_{n}}}\right)}$

and states that:

• If there exists a c such that ${\displaystyle \rho _{n}\geq c>0}$ for all n, then ${\displaystyle \sum _{k=1}^{\infty }a_{k}}$ converges
• If ${\displaystyle \rho _{n}\leq 0}$ for all n, and ${\displaystyle \sum _{k=1}^{\infty }1/D_{k}}$ diverges, then ${\displaystyle \sum _{k=1}^{\infty }a_{k}}$ diverges.

The proposed extension is written:

${\displaystyle {\overline {\rho }}_{n}=\left(D_{n}(1+\epsilon _{n})-D_{n+1}{\frac {a_{n+1}}{a_{n}}}\right)}$

where ε_n is a sequence of real numbers such that ${\displaystyle \lim _{n\to \infty }D_{n}\epsilon _{n}=0}$.

I think it can be shown that:

• If there exists a c such that ${\displaystyle {\overline {\rho }}_{n}\geq c>0}$ for all n, then there exists a c such that ${\displaystyle \rho _{n}\geq c>0}$ for all n and so ${\displaystyle \sum _{k=1}^{\infty }a_{k}}$ converges.
• If ${\displaystyle {\overline {\rho }}_{n}<0}$ for all n, and ${\displaystyle \sum _{k=1}^{\infty }1/D_{k}}$ diverges, then ${\displaystyle \rho _{n}<0}$ for all n and so ${\displaystyle \sum _{k=1}^{\infty }a_{k}}$ diverges.

The problem remains to either show that when ${\displaystyle {\overline {\rho }}_{n}=0}$ and ${\displaystyle \sum _{k=1}^{\infty }1/D_{k}}$ diverges, ${\displaystyle \sum _{k=1}^{\infty }a_{k}}$ diverges, or to find further restrictions on ε_n such that it does diverge, so that the convergence properties specified by ${\displaystyle {\overline {\rho }}_{n}}$ are the same as specified by ${\displaystyle \rho _{n}}$.

The above equation for ${\displaystyle {\overline {\rho }}_{n}=0}$ can be solved for a_n:

${\displaystyle a_{n}=a_{1}D_{1}{\frac {Q_{n}}{D_{n}}}}$

where Q_n is the product:

${\displaystyle Q_{n}=\Pi _{k=1}^{n-1}(1+\epsilon _{k})}$

and Q₁=1. We may take a_n and D_n to be unity, without loss of generality. So now we wish to find when

${\displaystyle \sum _{k=1}^{\infty }a_{k}=\sum _{k=1}^{\infty }{\frac {Q_{k}}{D_{k}}}}$

diverges. (Note that when ε_n = 0, the product Q_n = 1, and we recover the simple Kummer's test, and so ${\displaystyle \sum _{k=1}^{\infty }a_{k}}$ diverges since ${\displaystyle \sum _{k=1}^{\infty }1/D_{k}}$ diverges).

Abel's test applied to the above sequences states that if:

• ${\displaystyle \sum _{k=1}^{\infty }\epsilon _{k}Q_{k}}$ converges
• ${\displaystyle 1/\epsilon _{n}D_{n}}$ is monotone and bounded

then

${\displaystyle \sum _{k=1}^{\infty }{\frac {\epsilon _{k}Q_{k}}{\epsilon _{k}D_{k}}}=\sum _{k=1}^{\infty }a_{k}}$

coverges. In the present case, ${\displaystyle 1/\epsilon _{n}D_{n}}$ is NOT bounded since ε_nD_n -> 0, but I think Abel's theorem can be modified to prove that if

• ${\displaystyle \sum _{k=1}^{\infty }\epsilon _{k}Q_{k}}$ converges
• ${\displaystyle 1/\epsilon _{n}D_{n}}$ is unbounded

then

${\displaystyle \sum _{k=1}^{\infty }{\frac {\epsilon _{k}Q_{k}}{\epsilon _{k}D_{k}}}=\sum _{k=1}^{\infty }a_{k}}$

diverges.

So now we just need to deal with the first assumption (${\displaystyle \sum _{k=1}^{\infty }\epsilon _{k}Q_{k}}$ converges). It can be easily seen that

${\displaystyle Q_{n+1}-Q_{n}=(1+\epsilon _{n})Q_{n}-Q_{n}=\epsilon _{n}Q_{n}}$

so we want to look at the convergence of ${\displaystyle \sum _{k=1}^{\infty }(Q_{k+1}-Q_{k})}$ which equals Q-1 where Q is defined as:

${\displaystyle Q=\lim _{n\to \infty }\Pi _{k=1}^{n-1}(1+\epsilon _{k})}$

SO THE CONDITION THAT Q EXISTS is the restriction the ε_n must obey in order that ${\displaystyle \sum _{k=1}^{\infty }a_{k}}$ diverge for the case when ${\displaystyle {\overline {\rho }}_{n}=0}$.

According to Knopp^[1] (page 224, Theorem 9), if ${\displaystyle \sum _{k=1}^{\infty }|\epsilon _{k}|}$ converges, Q_n will converge to Q. Also according to Knopp (page 225 supplementary theorem), if ${\displaystyle \sum _{k=1}^{\infty }\epsilon _{k}^{2}}$ converges, Q_n will converge to Q.

So these are two restrictions on ε_n that will assure divergence of ${\displaystyle \sum _{k=1}^{\infty }a_{k}}$: ε_nD_n must converge to zero, and one or both of the above two conditions on ε_n apply. (Note these two conditions are sufficient, but not necessary: if they don't apply, that doesn't mean Q does not exist)