Word Count : 353

---

We are given the inequality: $$ 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{n}} \geq \sqrt{2(n-1)}, \quad \text{for } n \in \mathbb{N},\ n > 1. $$ We are to prove it manually. --- ### Approach: Use the integral comparison method. Let us denote: $$ S_n = \sum_{k=1}^{n} \frac{1}{\sqrt{k}}. $$ We aim to lower-bound this sum and show that: $$ S_n \geq \sqrt{2(n-1)}. $$ --- ### Step 1: Integral estimate ________ _____ __________ ______ ___ _________ ________ _____ ___.
________ __________ _____ _________ _________ _______.
___ _______ _________ ______ ____ _________ _______ _____.
_______ ______ ___ ________ __________ __________ ______.
_________ ______ ______ _____ _________ _______ ______ ______ ________.
______ ______ ___ ____ ______ __________ ____ _________ ________ ______.
______ _________ _______ _______ ___ _________ _________ _______.
_________ ___ ______ ________ _______ _____ __________ ________ ________ ______.
_____ _____ ________ _____ ________ _________ __________ ___ ________ ________ _______.
____ ____ ____ _______ _____ _____ _____ ____ __________ __________.
____ ___ _______ ___ ______ ________.
_____ _______ ___ ___ _____ _________ ________ _______ ______ ____.
___ _________ _____ _____ __________.
______ _______ _________ _____ ___ ____ __________ ________.
________ _______ __________ _______ __________ _____ _____ ________ ________ ______.
_________ ______ _________ _____ _________ _______ _____ ___ _____ __________ _________ _______.
___ __________ ______ ______ ________ ______ ________ ____ _____ __________ ______.
_______ _________ ______ ______ ________ __________ __________ __________ ____ _______.
____ ________ ________ ___ _______ _______ _________ _________ _________ ______.
_____ ___ _______ __________ ________ ___ _____ ____ _______ _________ _________ ____.
________ ____ _______ _____ _____ _________ ___.
_________ _______ _____ ______ __________ ________ ________ ______ ________.
_______ ______ ____ _____ _____ _________ _______ _____.
__________ _____ _____ ___ ___.
___ ______ _______ ___ __________ ____ ___ _______ _______ __________ _________ ________.
______ ________ ________ _______ ____ ___ __________ ___ ___ ___ _____ __________.
_______ ________ ___ ________ __________ ____ __________.
__________ ___ ___ ____ ____ ________.
__________ __________ _____ _______ ____ ________.
___ ________ ______ _____ ___ ____ _____ ____ _______ ___ ________.
___ _________ ___ __________ ________ ____ _______ _________ __________ __________ __________ ___.
_________ ____ _______ ________ _________ __________.
Get Full Answer on WhatsApp