Theorem 1.5. Consider the absolutely convergent series 56 S(x) = ao + a1x+ a2x² + + akxk + .. (1.246) .. where ak is a given function of k. Then S(x) can be expressed in the form ao xAao S(x) (1.247)...

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Extracted text: Theorem 1.5. Consider the absolutely convergent series 56 S(x) = ao + a1x+ a2x² + + akxk + .. (1.246) .. where ak is a given function of k. Then S(x) can be expressed in the form ao xAao S(x) (1.247) (1 – x)² (1 – æ)3 1 - x This result is known as Montmort's theorem on infinite summation. Note that if ak is a polynomial in k of degree n, then Amao will be zero for all m > n and thus a finite number of terms for the series S(x) will occur. 32 Difference Equations Proof. We have S(x) = ao + a1x + a2x² + •.. + akx* + ... (1+ xE + x² E² + · .. + x* Ek + ...· || (1 – xE)¯'ao = [1 – x(1+A)]¯'ao -1 1 1 1 - x (1.248) ao 1 - x 1 1+ 1- x ao 1- x (1 – x)2 xAao + ao 1- x (1 – x)² ' (1 – æ)