SUMMARY
The series \(\sum_{i=1}^{\infty}(\sqrt{n+1}-\sqrt{n})\) diverges. The transformation \(\sqrt{n+1}-\sqrt{n}=\frac{1}{\sqrt{n+1}+\sqrt{n}}\) is crucial for applying the comparison test. By comparing \(\frac{1}{\sqrt{n+1}+\sqrt{n}}\) with a known divergent series, one can establish the divergence of the original series. Analyzing the behavior of \(t_n \equiv \sqrt{n+1} - \sqrt{n}\) as \(n\) approaches infinity confirms this conclusion.
PREREQUISITES
- Understanding of series convergence and divergence
- Familiarity with the comparison test in calculus
- Knowledge of limits and asymptotic behavior
- Basic algebraic manipulation of square roots
NEXT STEPS
- Study the comparison test for series convergence in detail
- Learn about the behavior of sequences and series at infinity
- Explore the concept of asymptotic analysis in calculus
- Investigate other divergence tests, such as the ratio test and root test
USEFUL FOR
Students studying calculus, particularly those focusing on series and sequences, as well as educators looking for examples of divergence proofs.