Monotonicity of the ratio of two power series

[longshadow]

I'm thinking about the following function, which is a ratio of two finite power series. I'm trying to prove the monotonicity of this function, for arbitrary K.
\frac{\sum_{j=0}^k [(at)^j/j!]}{\sum_{j=0}^k [(bt)^j/j!]}, and a>b>0, t>0
I know that if k goes to infinity, the function becomes an exponential func. which is increasing in the domain t \in [0,\infty]. That is easy. But I'm wondering if this statement is true for arbitrary K.

Any thoughts?

 

[Greg Bernhardt]

Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

 

[platetheduke]

First off, the polynomials in your function are called partial sums rather than finite power series.

Let $p(t)=\sum_{j=0}^{k}\frac{t^j}{j!}$. Then your function is $\frac{p(at)}{p(bt)}$. We're going to replace $t$ with $\frac{t}{b}$ to get $\frac{p(ct)}{p(t)}$ with $c=a/b>1$. Now we will examine the simplified function by taking the derivative and letting $k>0$.

Note that $p^{\prime}(t)=p(t)-\frac{t^k}{k!}$.

$\frac{d}{dt}\frac{p(ct)}{p(t)}=\frac{cp^{\prime}(ct)p(t)-p(ct)p^{\prime}(t)}{p(t)^2}$

Since $p(t)$ has no negative coefficients, it is positive for $t>0$. So the denominator is positive. The numerator simplifies as

$cp^{\prime}(ct)[p^{\prime}(t)+\frac{t^k}{k!}]-[p^{\prime}(ct)+\frac{(ct)^k}{k!}]p^{\prime}(t)=(c-1)p^{\prime}(ct)p^{\prime}(t)+\frac{t^k}{k!}[cp^{\prime}(ct)-c^kp^{\prime}(t)]$

The first term is positive because $k>0$ and $c>1$. The second term is then of interest. I believe the second term is always negative for $t>0$ from looking at the powers of $c$. If you can show that the second term is never more negative than the first term is positive, then you are done.

Hope this helps. This problem bugs me in a good way. Thank you for sharing it. If I get more then I'll post.

 

[longshadow]

thank you for your reply! I'll see if I can get any new result.

Another way I've tried is:
1. Still working on the function in my original post, both the numerator(N) and the denominator(D) are positive, and when $t=0$, both N and D equal to one. So they have the same starting point.
2. When we take the $i$th derivative, we can see that for all $t$, the $i$th derivative of N is always larger than the $i$th derivative of D. This is hold for all $i <k$. So N always ''accelerates'' faster than D.
3. When $i>k$, both the $i$th derivatives of N and D are zero.

I feel that combining the above facts, we should be able to conclude that this function is monotonically increasing, but I cannot figure out how to prove it in a rigorous way.

Looking forward to more discussion with you!

 

[platetheduke]

With that line of thinking, consider this:
$\frac{p(ct)}{p(t)}=\frac{p^{\prime}(ct)+\frac{(ct)^k}{k!}}{p^{\prime}(t)+\frac{t^k}{k!}}$
It illustrates your point number 2. But I don't think it immediately finishes things.