A connection between the Root and Ratio Tests for Series in Calculus Courses

摘要: The Root Test and the Ratio for series $\sum_{n=0}^{\infty}a_n$ are usually discussed proved independently in Calculus courses. This can create students an impression that if one of these two tests is inconclusive because radius $\rho$ 1, then there a chance second test be useful. We suggest new approach to presentation tests, which eliminates this confusion. Suppose $\lim_{n\to\infty}\frac{|a_n|}{|a_{n+1}|}$ exists equal $\rho$. show implies $\lim_{n\to\infty}\sqrt[n]{|a_n|}=\rho$, limit root used verification convergence series. proof explains relation between Tests shows functional (and power particular) calculated with either methods.