Summation

Relevant wiki: Telescoping Series - Sum

$\large \begin{aligned} I & =\, \, \sum _{ k=1 }^{ 2016 }{ \frac { 1 }{ \sqrt { k+1 } +\sqrt { k } } } & \text{Question} \\ & =\, \, \sum _{ k=1 }^{ 2016 }{ \frac { 1 }{ \sqrt { k+1 } +\sqrt { k } } \left( \frac { \sqrt { k+1 } -\sqrt { k } }{ \sqrt { k+1 } -\sqrt { k } } \right) } & \text{Rationalize the denominator} \\ & =\, \, \sum _{ k=1 }^{ 2016 }{ \frac { 1\left( \sqrt { k+1 } -\sqrt { k } \right) }{ \left( \sqrt { k+1 } +\sqrt { k } \right) \left( \sqrt { k+1 } -\sqrt { k } \right) } } & \text{Multiply fractions} \\ & =\, \, \sum _{ k=1 }^{ 2016 }{ \frac { \sqrt { k+1 } -\sqrt { k } }{ \left( \sqrt { k+1 } +\sqrt { k } \right) \left( \sqrt { k+1 } -\sqrt { k } \right) } } & \color{#BBBBBB}{1 \times a = a} \\ & =\, \, \sum _{ k=1 }^{ 2016 }{ \frac { \sqrt { k+1 } -\sqrt { k } }{ \left( \sqrt { k+1 } \right) ^{ 2 }-\left( \sqrt { k } \right) ^{ 2 } } } & \color{#BBBBBB}{(a+b)(a-b)=a^{2}-b^{2}} \\ & =\, \, \sum _{ k=1 }^{ 2016 }{ \frac { \sqrt { k+1 } -\sqrt { k } }{ { k }+1-k } } & \text{Simplify and cancel} \\ & =\, \, \sum _{ k=1 }^{ 2016 }{ { \sqrt { k+1 } -\sqrt { k } } } & \text{Simplify} \\ & =\, \, \, \, \left( \sqrt { 2017 } -\sqrt { 2016 } \right) +\left( \sqrt { 2016 } -\sqrt { 2015 } \right) +\dots +\left( \sqrt { 3 } -\sqrt { 2 } \right) +\left( \sqrt { 2 } -\sqrt { 1 } \right) & \\ & =\, \, \, \, \sqrt { 2017 } -\sqrt { 2016 } + \sqrt { 2016 } -\sqrt { 2015 } +\dots + \sqrt { 3 } -\sqrt { 2 } + \sqrt { 2 } -\sqrt { 1 } & \text{Remove brackets and cancel} \\ & =\, \, \, \, \sqrt { 2017 } -1 & \text{Simplify} \\ & \approx \, \, \, 43.9 & \\ \left\lceil I \right\rceil \, \, & =\, \, \, \, \, \left\lceil 43.9 \right\rceil & \text{Use ceiling function} \\ & =\, \, \, \, \, \boxed { 44 } & \end{aligned}$

Relevant wiki: Telescoping Series - Sum

This is a rather simple one. $\frac{1}{\sqrt{k+1}+\sqrt{k}}=\frac{\sqrt{k+1}-\sqrt{k}}{({\sqrt{k+1})}^{2}-({\sqrt{k}})^{2}} = \sqrt{k+1}-\sqrt{k}$ Now, $I=\sum_{k=1}^{2016}\frac{1}{\sqrt{k+1}+\sqrt{k}}=\sqrt{2017}-\sqrt{2016}+\sqrt{2016}-\sqrt{2015}....+\sqrt{2}-1=\sqrt{2017}-1$ This can be calculate to be approximately $43.9$ , which the ceiling function makes $44$ .