Nov 16, 2022 · In this section we will discuss using the Comparison Test and Limit Comparison Tests to determine if an infinite series converges or diverges. In order to use either test the terms of the infinite series must be positive. Proofs for both tests are also given.

Theorem: Limit Comparison Test. Let [latex]{a}_{n},{b}_{n}\ge 0[/latex] for all [latex]n\ge 1[/latex]. If [latex]\underset{n\to \infty }{\text{lim}}\frac{{a}_{n}}{{b}_{n}}=L\ne 0[/latex], then [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] and [latex]\displaystyle\sum _{n=1}^{\infty }{b}_{n}[/latex] both converge or both diverge.

Oct 17, 2023 · The first key is to choose a test series that you know converges or diverges AND that will help you get a finite, positive limit. Idea 1: If you have polynomials in both the numerator and denominator of a fraction, drop all terms except for the highest power terms (in both parts) and simplify. Drop any constants.

Jan 23, 2025 · The numerator is equal to \(1\) when \(n\) is odd and \(0\) when \(n\) is even, so the series can be rewritten \(\displaystyle \sum^∞_{n=1}\frac{1}{2n+1},\) which diverges by limit comparison with the harmonic series.

The limit comparison test is an easy way to compare the limit of the terms of one series with the limit of terms of a known series to check for convergence or divergence. It may be one of the most useful tests for convergence.

Limit Comparison Test. The limit comparison test is similar to the comparison test in that you use another series to show the convergence or divergence of a desired series. Suppose we have two series

Nov 16, 2022 · \[\begin{align*}c & = \mathop {\lim }\limits_{n \to \infty } \frac{{{a_n}}}{{{b_n}}} = \mathop {\lim }\limits_{n \to \infty } \left[ {{a_n}\frac{1}{{{b_n}}}} \right] = \mathop {\lim }\limits_{n \to \infty } \left[ {\frac{{\sqrt {2{n^2} + 4n + 1} }}{{{n^3} + 9}}\,\,\frac{{{n^2}}}{{\sqrt 2 }}} \right]\\ & = \mathop {\lim }\limits_{n \to \infty ...

Use the Limit Comparison Test to determine the convergence or divergence of the series \begin{equation*} \sum \frac{3k^2+1}{5k^4+2k+2}\text{.} \end{equation*} by comparing it to the series \(\sum \frac{1}{k^2}\text{.}\)

The Limit Comparison Test is a good test to try when a basic comparison does not work (as in Example 3 on the previous slide). The idea of this test is that if the limit of a ratio of sequences is 0, then the denominator grew much faster than the numerator.

Dec 29, 2024 · Learn how to use the limit comparison test with our step-by-step guide. Master series convergence testing, understand key principles, and apply this essential calculus technique effectively. Explore detailed examples, common pitfalls, and expert tips to enhance your problem-solving skills. Perfect for students and educators seeking …