Simple and interesting inequality involving the Lerch transcendent function

Question

I’m trying to prove this general inequality related to the Lerch transcendent function: $$ \ell \cdot \left(\Phi(\frac{1}{\ell + \frac{c}{\sqrt{c^2+1}}}, \frac{1}{2}, c^2) - \frac{1}{c}\right) \geq \frac{1}{\sqrt{c^2+1}} \ , $$ for every $\ell + \frac{c}{\sqrt{c^2+1}} > 1$ and every $c \in \mathbb{N}$. Let’s write it for $c=1$ in another way: $$ \ell \cdot \left(\sum_{n=1}^\infty \frac{1}{\sqrt{n+1}} \cdot \frac{1}{(\ell + \frac{1}{\sqrt{2}})^n}\right) \geq \frac{1}{\sqrt{2}} \ . $$ I have numerically verified that this inequality holds. Moreover, I numerically observed that the LHS is decreasing in $\ell$, which could prove the inequality by taking $\ell \to +\infty$. However, I can neither prove the monotonicity in $\ell$ nor the inequality itself. Any ideas how to solve this? I have tried numerous ways, e.g., truncating the infinite sum by considering the first few terms wouldn’t work for every $c$ (even for $c=1$ we need 6 terms which gives a nasty expression).