How to simplify this summation, or express as integral? $\sum_{t=-\infty}^\infty\frac1{\sqrt{(t+ax)^2+4}}$

Question

How to simplify this summation, or express as integral? $$f(x)=\sum_{t=-\infty}^\infty\frac1{\sqrt{(t+ax)^2+4}}$$ $a$ is a constant, say, 24. $$f(x)=\sum_{t=-\infty}^\infty\frac1{\sqrt{(t+24x)^2+4}}$$

Answer 1

Just to make an answer from sos440's comment: From the integral test you get:

$$\sum_{t=1}^\infty\frac1{\sqrt{(t+24x)^2+4}} > \int_1^\infty \frac1{\sqrt{(t+24x)^2+4}} dt = \infty$$

because $\frac1{\sqrt{(t+24x)^2+4}} \approx \frac 1x$ for $x\rightarrow\infty$ and $\int_1^\infty \frac 1x dx$ diverges. Because each summand of $f(x)$ is positive, also $$\sum_{t=-\infty}^\infty\frac1{\sqrt{(t+24x)^2+4}} = \infty$$.