Following on from my question here about the integral equation
$$y(x)=1+\int^{x}_{0}(\tanh s)y(s)ds$$
we now look to appeal to Picard's Theorem.
Let $\{y_n\}_{n \geq 0}$ be the sequence of Picard Approximations for which
$$y_0(x)=1$$
$$y_{n+1}=1+\int^{x}_{0}\tanh(s)y_n(s)ds \quad \quad (n \geq 0)$$
We're asked to prove that
$$y_n(x)=\sum_{k=0}^{n} \frac{1}{k!} (\log \cosh x)^k$$ and that $y_n \longrightarrow y$ as $n \longrightarrow \infty$.
We know the solution $y$ to be the function $y:=\cosh(x)$.
Any help with determining how this works would be very appreciated. Regards and best, as always. MM