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Bounding Summations

\( \sum_{k=0}^n x^k \leq \sum_{k=0}^{\infty} x^k = \frac{1}{1-x} \; {\rm
if} \; \vert x\vert < 1 \)

\( \sum_{k=1}^n (k-1) = \sum_{k=0}^{n-1} k \)

\( \sum_{k=1}^n (n-k) = \sum_{k=0}^{n-1} k \)

\( \sum_{k=1}^n a_k \leq n a_{max} \)



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