Compare Expression (Normal)

$\frac { S }{ 2 } =\frac { 1 }{ 2\sqrt { 1 } } +\frac { 1 }{ 2\sqrt { 2 } } +\frac { 1 }{ 2\sqrt { 3 } } +...+\frac { 1 }{ 2\sqrt { 2023 } } +\frac { 1 }{ 2\sqrt { 2024 } } \\ Have:\quad 0<2\sqrt { n } <\sqrt { n } +\sqrt { n+1 } \Rightarrow \frac { 1 }{ 2\sqrt { n } } >\frac { 1 }{ \sqrt { n } +\sqrt { n+1 } } =-\sqrt { n } +\sqrt { n+1 } \\ \Rightarrow \frac { S }{ 2 } >-\sqrt { 1 } +\sqrt { 2 } -\sqrt { 2 } +\sqrt { 3 } -...-\sqrt { 2023 } +\sqrt { 2024 } -\sqrt { 2024 } +\sqrt { 2025 } =-\sqrt { 1 } +\sqrt { 2025 } =-1+45=44\\ \Rightarrow S>88.$

Consider the left Riemann sum of $f(x)=\frac{1}{\sqrt{x}}$ on $[1,2025]$ with mesh size 1. Now $\int_{1}^{2025}x^{-1/2}dx=2[x^{1/2}]_{1}^{2025}=2(45-1)=88<\sum_{k=1}^{2024}\frac{1}{\sqrt{k}}=S$