# Does the following sequence $\{g_n\}$ converge?

Consider a function sequence $$\{f_n(t)\}$$ ($$n\in\mathbb{N}^+$$) defined on the interval $$(\frac{1}{2},1)$$, where $$\begin{eqnarray}\label{eqn:constraint1} f_n(t)=\frac{\exp\left(n\left(\log R(h_t) - th_t\right)\right)}{h_t\sigma(h_t)\sqrt{2\pi n}}\left(1+o\left(1\right)\right) \end{eqnarray}$$ as $$n\to \infty$$ uniformly in $$t$$ in any closed subinterval of the interval $$(\frac{1}{2},1)$$, $$\begin{eqnarray} R(h)=\frac{e^h-1}{h}, \end{eqnarray}$$ $$\begin{eqnarray} m(h)=\frac{R'(h)}{R(h)}=\frac{e^h}{e^h-1}-\frac{1}{h}, \end{eqnarray}$$ $$\begin{eqnarray} \sigma(h)=m'(h), \end{eqnarray}$$ and $$h_t$$ is the positive real root of the equation $$m(h)=t$$.

Then we study the following equation of $$g_n$$ $$\begin{eqnarray} c=g_n+\frac{\int_{g_n}^1f_n(t)~dt}{f_n(g_n)},~\frac{1}{2}< g_n<1, \end{eqnarray}$$ where the constant number $$\frac{1}{2}< c<1$$.

Try to prove:

1. the convergence of the sequence $$\{g_n\}$$ and compute $$A=\lim_{n\to \infty}g_n$$;
2. $$\lim_{n\to \infty}n(g_n-A)=0$$.
• series or sequence? – Zhou Aug 17 at 3:56
• Sorry for this typo. I have corrected it. – Ryan Chen Aug 17 at 7:13
• did you try to code it up? for $o(1)$ just try different things. – Sina Baghal Aug 18 at 5:22
• Can you tell where this problem comes from please. Sometimes a problem is more easily solved in its original form and then follows by applying an appropriate continuous map. – Dieter Kadelka Aug 18 at 8:31