Sorry, let me pose the full problem and I think you can help me. For any positive integer $x$, let $t(x)$ be a real number with a priori is such that $t(x)>1$ and $t(x)$ tends to $1$ as $x\to \infty$.

Now, denote $q=q(x)=e^{-2\pi t(x)}$ and $q_0=e^{-2\pi}$.

I have that $$ j(q)-j(q_0)=e^{-2\pi t(x)}-e^{-2\pi}+\sum_{k\geq 1}c_k(e^{-2k\pi t(x)}-e^{-2k\pi}). $$

Thus, all I wish is an explicit lower bound for $j(q)-j(q_0)$ in terms of $t(x)-1$.

Sorry for the previous post. I thank for all suggestions and ideas for bounding $(d_k)$ (which was one of my ideas to get the previous lower bound).

Any hint?