Numerator...Denominator...Terminator!

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Consider the following sequence -

\[\displaystyle \frac{1}{2}, \frac{5}{3}, \frac{11}{8}, \frac{27}{19} \cdots\]

Now, if \({T}_{n}\) is the nth term of this sequence and \(N({T}_{n})\) is the numerator of the term and \(D({T}_{n})\) is the denominator of the term, then find \[\displaystyle \text{digit sum} \left (\dfrac{N({T}_{17}) + N({T}_{29}) - D({T}_{17}) - D({T}_{29})}{{(\displaystyle \lim_{n \rightarrow \infty}({T}_{n}))}^{2}} \right)\]

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