The limit of a sequence reveals its long-term behavior. In this article, we will provide a step-by-step guide to determining the limit of a sequence. Step 1: Does the Sequence Converge? The first step to finding the limit of a sequence is to figure out if it converges or not. For example, the sequence \(a_n…
Read articleThe Alternating Series Test is a fundamental tool in calculus used to determine whether certain infinite series converge. This test specifically applies to series that alternate in sign, meaning the terms switch between positive and negative. In this article, we’ll discuss the test in detail and walk through three examples of increasing difficulty. The Alternating…
Read articleYou might have been searching for “Alternate Series Test,” but the proper term is actually “Alternating Series Test.” This test is a useful tool in calculus for determining the convergence of certain types of infinite series, specifically those that alternate in sign. In this article, we’ll explore what the Alternating Series Test is and how…
Read articleIn this article, we will quickly prove that the harmonic series, \(\sum_{n=1}^{\infty} \frac{1}{n}\), diverges. Does 1/n converge? Proof We will split the terms of the series as follows: \[ 1 = 1 \] \[ \frac{1}{2} = \frac{1}{2} \] \[ \frac{1}{3} + \frac{1}{4} > \frac{1}{4} + \frac{1}{4} = \frac{1}{2} \] \[ \frac{1}{5} + \frac{1}{6} + \frac{1}{7}…
Read articleSeries are one of the most frustrating topics in calculus. They can be very confusing, and it can be very hard to determine if a series converges or not. If that weren’t enough, there are two types of convergence – absolute and conditional. This article will take you through what conditional convergence is and how…
Read articleThe Ratio Test is a very nice tool to use for determining whether or not a series converges, but it can get a bit complicated. Here it is: If we have a series \(\sum a_n\), let there be a number \(L = \lim_{n\to\infty} \left|\dfrac{a_{n+1}}{a_n}\right|\). – If \(L < 1\), the series converges absolutely. – If…
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