(The root test): Let an ≥0. (i) If lim supn →∞ √(an)<1, show that ∑n=1^∞ an<∞. (iii) Find exampl

Step 1: Understand the root test for convergence of series.u000AThe root test states that for a seri

(The root test): Let an ≥0. (i) If lim supn →∞ √(an)<1, show...

(The root test): Let $a_n \geq 0$.
(i) If lim $\sup _{n \rightarrow \infty} \sqrt[n]{a_n}<1$, show that $\sum_{n=1}^{\infty} a_n<\infty$.
(iii) Find examples of both a convergent and a divergent series having $\lim _{n \rightarrow \infty} \sqrt[n]{a_n}=1$

Key Concepts

Convergence of Infinite Series

Convergence of an infinite series refers to the behavior where the sum of its terms approaches a finite limit. Various tests, such as the root test and ratio test, are employed to decide if an infinite series converges or diverges. Understanding these tests and their conditions is crucial for analyzing series in mathematical analysis.

Borderline Cases in Convergence Tests

Borderline cases occur when the criterion used to test for convergence, such as the limit of the nth root of the terms, equals one. In these scenarios, the usual tests like the root test become inconclusive, and additional investigation or alternate convergence tests are necessary. Highlighting and constructing examples in such cases is important for deeper insights into series behavior and the limitations of standard tests.

Root Test

The root test is a criterion used to determine the convergence or divergence of an infinite series by examining the nth root of its terms. If the limit superior of the nth roots is less than one, the series converges; if it is greater than one, the series diverges; and if it equals one, the test is inconclusive. This method is especially useful for series with terms raised to the power n or involving exponential growth or decay.

Limit Superior (lim sup)

The limit superior of a sequence, often denoted as lim sup, is a concept that captures the largest accumulation point or the long?term upper bound of a sequence. In the context of the root test, it helps in determining the overall limiting behavior of the nth roots of the terms of a series, guiding the convergence analysis when the sequence does not have a straightforward limit.

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