b. Let \( r \) and \( s \) be any real numbers. By multiplying the series expansions of \( (1+t)^{r} \) and \( (1+t)^{s} \), prove that \[ \left(\begin{array}{c} r+s \\ n \end{array}\right)=\sum_{k=0}^{n}\left(\begin{array}{l} r \\ k \end{array}\right)\left(\begin{array}{c} s \\ n-k \end{array}\right) . \]
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These expansions are given by: \[ (1+t)^r = \sum_{k=0}^{\infty} \left(\begin{array}{c} r \\ k \end{array}\right) t^k \] \[ (1+t)^s = \sum_{k=0}^{\infty} \left(\begin{array}{c} s \\ k \end{array}\right) t^k \] where \( \left(\begin{array}{c} r \\ k Show more…
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