TIFR-2020-Maths-A: 9

Question

What is the greatest integer less than or equal to

$$\sum_{n=1}^{9999}\frac{1}{\sqrt[4]{n}}?$$

• A. $1332$
• B. $1352$
• C. $1372$
• D. $1392$

Answer 1

$\text {By using}$ Approximation by integrals

$\text{Since, $n^{-1/4}$ is a monotonically decreasing function}, $

$\int_{1}^{10000}n^{-1/4}\;dn \leqslant \sum_{n=1}^{9999}n^{-1/4} \leqslant \int_{0}^{9999}n^{-1/4}\;dn$

$1332 \leqslant \sum_{n=1}^{9999}n^{-1/4} \leqslant 1333.23$

$\text{Hence, Answer: (a)}$