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 articleYou may remember from your math class that \(\displaystyle\int \cos x \, dx = \sin x + C\), but what happens if it is integral of -cos x? Negative sign is a multiplication by (-1) To get from $\cos x$ to $-\cos x$, we need to multiply it by a negative one: \[ \begin{align*} (-1)…
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 articleIn calculus, finding the integral of trigonometric functions is a common task. One of the most fundamental integrals is the integral of \(\sin(x)\). In this short article, we will explore how to find the integral of \(\sin(x)\). Finding the Integral The integral of \(\sin(x)\) can be found using basic integration rules. We know that the…
Read articleIntegrals and derivatives of trigonometric functions can be tricky. You may have been told in your math class that $\dfrac{d}{dx} \sin(x) = \cos(x)$, but not given a good explanation for it. In this article, we will show you with proof how to find the derivatives and integrals of $\cos x$ and $\sin x$ functions. Derivatives…
Read articleTo find the integral of x 2, written as \(\displaystyle \int x^2 \, dx\), we use the Power Rule for integration. The Power Rule for differentiation states: \[ \dfrac{d}{dx}(x^n) = n \cdot x^{n-1} \] To integrate, we reverse this process with what’s called the Reverse Power Rule: \[ \displaystyle \int x^n \, dx = \frac{x^{n+1}}{n+1}…
Read articleHave you ever wondered how to find the integral of \(\sqrt{x}\)? It’s actually quite simple once you know the power rule! Let’s start by rewriting \(\sqrt{x}\) in a way that’s easier to work with. We know that: \[ \sqrt{x} = x^{\frac{1}{2}} \] Now, we can use the power rule for integration. The power rule says…
Read articleIntroduction In calculus, we often need to find the limit of a function as it approaches a certain value. A common scenario involves the limit of a ratio of two functions, which we call rational functions. The general form looks like this: $$\lim_{x \to a} \frac{f(x)}{g(x)}$$ This expression asks: “What value does the fraction approach…
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