Show that .b) Let . Show that

Question

Show that $1 + \frac{1}{\sqrt{2}} + \dots + \frac{1}{\sqrt{n}} > 2\sqrt{n + 1} - 2, \forall n \in \mathbb{N}$ .
b) Let $a, b > 0, a + b = 1, n > 1$ . Show that $\left(a + \frac{1}{a}\right)^n + \left(b + \frac{1}{b}\right)^n \ge \frac{5^n}{2^{n-1}}$

09 Jan 2026

Answer :

Word Count : 115

For (a), consider the function (f(x)=\frac{1}{\sqrt{x}}), which is positive and decreasing on ([1,\infty)). For each integer (k\ge1), [ \int_k^{k+1}\frac{dx}{\sqrt{x}}<\frac{1}{\sqrt{k}}. ] Hence, [ \sum_{k=1}^n\frac{1}{\sqrt{k}}> \int_1^{n+1}\frac{dx}{\sqrt{x}}. ] Evaluating the integral, _______ _____ ______ _____ ________.
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