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I am thinking about applying Tannery's theorem to some generalised hypergeometric functions, which seems to be a standard method to derive various formulæ. For example, \begin{eqnarray} \lim_{n\to+\infty}{}_3F_2\Big( \begin{matrix} a,b,c-n, \\ a-b+1,a+c+1+n \end{matrix} ;1\Big), \end{eqnarray} To apply Tannery's theorem, we need to find a sequence $(d_k)$ such that \begin{eqnarray} |a_k| \leq d_k, \quad \sum_{k=0}^\infty d_k < +\infty \end{eqnarray} where $a_k$ is defined as the summand of the ${}_3F_2$ above, \begin{eqnarray} a_k:=\frac{(a)_k(b)_k(c-n)_k}{k!(a-b+1)_k(a+c+1+n)_k} \end{eqnarray} I would like to know one or more concrete examples of such $(d_k)$ with details about why they satisfy the conditions above.

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