AP EAMCET · Maths · Sequences and Series
For all \(\mathrm{n} \in \mathrm{N}, 1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\ldots+\frac{1}{\sqrt{n}}\) is
- A \(>n\)
- B \( < \sqrt{n}\)
- C \(\leq \sqrt{n}\)
- D \(\geq \sqrt{n}\)
Answer & Solution
Correct Answer
(D) \(\geq \sqrt{n}\)
Step-by-step Solution
Detailed explanation
For \( \mathrm{n}=1 \): \( 1 \geq \sqrt{1} \), which is true. Assume \( 1+\frac{1}{\sqrt{2}}+\ldots+\frac{1}{\sqrt{\mathrm{k}}} \geq \sqrt{\mathrm{k}} \) for some \( \mathrm{k} \in \mathrm{N} \). We prove for \( \mathrm{k}+1 \):…
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