Determine the convergence of ∑n=1∞1n+n+n\sum_{n=1}^{\infty} \frac{1}{\sqrt{n + \sqrt{n + \sqrt{n}}}}∑n=1∞n+n+n1.
Which statement is correct?
By direct comparison with ∑1n\sum \frac{1}{\sqrt{n}}∑n1: since the denominator is larger, the series diverges.
By bounding n+n+n≥n\sqrt{n + \sqrt{n + \sqrt{n}}} \geq nn+n+n≥n, the series converges.
By asymptotic analysis: n+n+n∼n\sqrt{n + \sqrt{n + \sqrt{n}}} \sim \sqrt{n}n+n+n∼n, so Limit Comparison Test with ∑1n\sum \frac{1}{\sqrt{n}}∑n1 shows divergence.
By integral test: since ∫1∞dxx\int_1^{\infty} \frac{dx}{\sqrt{x}}∫1∞xdx diverges, the series diverges.