I am attempting to code up an equation that includes an infinite sum of cosine and hyperbolic cosine functions, namely:
$$ \sum_{m=0}^{\infty} \frac{ \cos[(2m+1)\pi x/s] \cosh[(2m+1)\pi z/s] } {(2m+1)^2 \cosh[(2m+1)\pi z_0/s]} $$
According to the research paper from which this equation is taken, this infinite sum is convergent. However, my understanding is that, as $m$ approaches infinity, so will the hyperbolic cosine terms, as will the $(2m+1)^2$ term in the denominator.
Is this infinite series truly convergent? And if so, under which conditions?
Note that $s$ is represents the maximum value of $x$ and $z_0$ is the maximum value of $z$. Therefore the ratio $x/s$ will range from 0 to 1. $s$ will also always be considerably larger (i.e. typically an order of magnitude or more) than either $z$ or $z_0$, so $z/s$ and $z_0/s$ will both be less than 1.
For those interested, the paper in question is Tóth, J. (1962). A theory of groundwater motion in small drainage basins in central Alberta, Canada. J. Geophys. Res, 67(11), 4375-4387.
For large $y$, $\cosh{y} \sim e^{|y|}/2$. Therefore the ratio of hyperbolic cosines is asymptotic to $$ e^{-(2m+1)\pi(z_0-|z|)/s}, $$ which goes to zero exponentially fast when $|z|<z$. Moreover, it is easy to check the ratio is less than or equal to $1$ for all $m$. The cosine is bounded by one, so does not affect convergence, and the $\sum_{m \geqslant 0} 1/(2m+1)^2$ also converges, so the sum is bounded by $$ \sum_{m \geqslant 0} \frac{1}{(2m+1)^2} \cdot 1 < \infty. $$