Determining Absolute Convergence Using Ratio or Root Test

Using the ratio or root test, determine if the series [{ "type" : "katexPlugin", "text": "\sum a_n", "inline": true }]', is absolutely convergent.

When faced with an infinite series, determining its convergence is a fundamental task in calculus and analysis. The convergence of series is a vital concept because it tells us if the sum of infinitely many terms can reach a finite value. One common technique to determine this property is through the use of the ratio test and the root test, two powerful tools designed for this purpose. The ratio test analyzes the limit of the ratio of successive terms of the series. If the limit of the absolute value of consecutive term ratios is less than one, the series is absolutely convergent. On the other hand, if it is more than one or diverges to infinity, the series is divergent. Meanwhile, the root test checks the limit of the nth root of the absolute value of the terms. Similar to the ratio test, if this limit is less than one, absolute convergence is achieved; if more than one, the series diverges. Indeterminate results occur when these limits equal exactly one, and those cases require alternative techniques or tests for convergence.