Consider the function $\Lambda(k)$, which is an even function in $k$, and it has following conditions For small values of $k$ $$\Lambda(k) \sim x_0^2$$ and for large values of $k$, the behaviour of the function $$\Lambda(k) \sim \frac{1}{k^2}$$
In addition to these conditions $\Lambda(k)$ is an even function in k and $\Lambda(k) > 0 \;\;(\forall k \in \mathbb{R})$
From these conditions, we can possibly write series expansions about $k=0$ and $k=\infty$, as follows,
at $k=0$, series can be written as:
$$\Lambda(k) = x_0^2\left( 1 + \sum_{n=1}^\infty c_n (kx_0)^{2n}\right)$$
On the other hand, at $k\rightarrow\infty$, the series can be written as, $$\Lambda(k) \sim x_0^2 \left( \frac{1}{(kx_0)^2} +\frac{b_1}{(kx_0)^4} + O\left[k^{-6}\right] \right)$$
The question is:
Can a general even function $\Lambda(k)$ be defined that satisfies these conditions, namely the limits as $k\to 0$ and $k\to \infty$?
A particular example is $$ \Lambda(k) = \frac{x_0^2}{1+(k x_0)^2} $$ which can be written in the series form for $|k| < 1$ with $c_n = (-1)^n$.
PS: An additional note is that $\Lambda(k)$ is a Fourier Transform of a function $\Lambda(x)$.