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P-Series Test

One of the most useful tests for the convergence of series is the p-series test. It is a comparison test that is deeply related to the convergence of a class of improper Riemann integrals. P-series Test proof To prove the p-series test, we need some preliminaries on the relationship between improper integrals and the series. Integral test for series Theorem: Let $f:[0,+infty)$...

Root test for series

Several series convergence tests are available in the mathematics course. Here we prove the root test for series. It is so called because such a test uses the nth root of the general term of the series. Generalities on root test for series and examples Here we use series with positive terms. This series $sum_{n=0}^{+infty}u_n$ with $u_nge 0$ for any positive...

Geometric Series

One of the best-known series is the geometric series. It is called by such a name because its general term is a geometric sequence. Here on this page, we give more details about the convergence of this series. This series has several applications in mathematical analysis and the probability theorem. There is a probability law called the geometric distribution,...

Harmonic series

On this page, we prove that the harmonic series is a divergent series. In fact, this is one of the best-known and classic examples of the infinite series course. Why does the harmonic series diverge We start by proving classical results on sequences. If a sequence $(u_n)_n$ is increasing and has no upper bound. Then $lim_{nto+infty}u_n=+infty$. In fact, by assumption for...

Limits at infinity

For several mathematical models, one generally needs to study the limits at infinity of the solution. It is a kind of stability of systems. We then recall this concept and give some examples. We already discussed the limits at the finite points of functions. Now we discuss the case of infinite points. Definitions and properties of the limits at infinity Let $a,$...

Uniformly continuous function

Any uniformly continuous function is actually a continuous function. However, as we will see below, the converse is not true. This important class of functions satisfies useful properties. In this article, we will discuss all these facts in detail. The uniform continuity of a function In some mathematical problems, continuity is not enough to make a decision. In fact,...

Functions of one variable

We provide all necessary properties of functions of one variable such as limit at a given point, continuity, and differentiability. Such functions are defined over a domain of the set of real numbers. Continuity of functions of one variable We denote by $mathbb{R}$ the set of real numbers. If $f$ is a real function, then we denote by $mathscr{D}_f$ the...

Mean value theorem

One of the most fundamental theorems in mathematical analysis is the mean value theorem. Geometrically, the theorem says that somewhere between points A and B on a differentiable curve there is at least one tangent line parallel to the secant line AB. Let's discover together this great theorem and give it some applications. Here will use the concept of differential...

Ratio test for series

One of the most practical convergence tests for series is the ratio test. Just a limit of a ratio defined by the general term of the series can tell us if the series converges or diverges. We mention that in the French mathematical school this test is called the d'Alembert rule. This is because it was founded by the...

Convergent series examples

The main purpose of this article is to provide examples of convergent series. We also speak of divergent series. Before that, we first give a concise summary of the properties of the series and recall the convergence criteria of the series with the proofs. In fact, to fully understand the contents of this page, information on convergence sequences is...