TIFR ECE 2022 | Question: 15

Question

Consider the difference below for $m \geq 5$:

\[\sum_{n=1}^{m-1} \frac{1}{(1+n)^{2}}-\int_{x=1}^{m} \frac{1}{(1+x)^{2}} d x .\]

Which statement about the difference is $\text{TRUE}?$

• A. It is positive for infinitely many $m \geq 5$ and negative for infinitely many $m \geq 5$
• B. It is positive for all $m \geq 5,$ and is increasing as $m$ increases 
• C. It is negative for finitely many $m \geq 5$ and is positive for infinitely many $m$
• D. It is positive for all $m \geq 5,$ and is decreasing as $m$ increases
• E. It is negative for all $m \geq 5$