Enveloping expansion for series 33

Let series #33 in Table have $t=-1$, and let $\mathcal{R}_n$ and $\mathcal{F}_n$ be the quantities associated with this series. Let $c_j$ be the coefficients defined by Theorem. Enveloping-expansion conjecture. Theorem is also valid for series #33, and, for every $L\in\mathbb N$, its expansion satisfies

$$\sum_{j=0}^{4L-1}\frac{c_j}{n^j} < \frac{\mathcal{R}_n}{\mathcal{F}_n} < \sum_{j=0}^{4L+1}\frac{c_j}{n^j}.$$

The conjecture concerns the exceptional case $t=-1$, which lies outside the proof's hypothesis $|t|<1$. The displayed inequalities assert an enveloping expansion and give numerical control of the truncated asymptotic series for this particular hypergeometric series.