SUMMARY
The series \(\sum^{\infty}_{n=1} \frac{1}{\sqrt{n+1}+\sqrt{n}}\) is determined to be divergent by comparing it to the p-series \(\sum^{\infty}_{n=1} \frac{1}{\sqrt{n}}\), which is known to diverge. The rationalization approach suggested in the discussion is not necessary; instead, evaluating the limit of the function as \(n\) approaches infinity reveals that it behaves like \(\frac{1}{2\sqrt{n}}\). This confirms the divergence of the original series.
PREREQUISITES
- Understanding of series convergence and divergence
- Familiarity with p-series and their properties
- Knowledge of limits and asymptotic behavior
- Basic algebraic manipulation techniques, including rationalization
NEXT STEPS
- Study the properties of p-series and their convergence criteria
- Learn about the Limit Comparison Test for series
- Explore techniques for evaluating limits of series as \(n\) approaches infinity
- Review rationalization methods and their appropriate applications in series analysis
USEFUL FOR
Students studying calculus, particularly those focusing on series convergence, mathematicians, and educators looking to enhance their understanding of series analysis techniques.