Home > Problems > Another short quiz.

Another short quiz.

September 30, 2009 Leave a comment Go to comments

Show that
\sum_{i=1}^k \frac{k-i}{\sqrt{i}+\sqrt{i+1}}=\sqrt{1}+\sqrt{2}+\sqrt{3}+\dots+\sqrt{k-1}+\sqrt{k} .
Hence, proof
\sum_{i=1}^k \frac{k-i}{\sqrt{i}+\sqrt{i+1}}\leq k \sqrt{\frac{k+1}{2}}.
Note: The second part is actually a practice of the Cauchy-Schwarz inequality, which in general, for a_n , b_n \, \in \textbf{R} \, \text{for} \, 1 \leq n \leq i :
\left({a_1}^2+{a_2}^2+\dots+{a_i}^2\right)\left({b_1}^2+{b_2}^2+\dots+{b_i}^2\right)\geq \left(a_1 b_1 + a_2 b_2 +\dots + a_i b_i\right)^2

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