Lecture

Geometric Series: Properties and Applications

Graph Chat

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.

Description

This lecture covers the properties and applications of geometric series, including the definition of logarithmic functions, their graphical representation, and the relationship between exponential and logarithmic functions. It also explores the area under curves, theorems related to geometric series, and the Mercator series.

This video is available exclusively on Mediaspace for a restricted audience. Please log in to MediaSpace to access it if you have the necessary permissions.

Watch on Mediaspace

Official source

Related concepts (32)

Logarithm

In mathematics, the logarithm is the inverse function to exponentiation. That means that the logarithm of a number x to the base b is the exponent to which b must be raised to produce x. For example, since 1000 = 103, the logarithm base 10 of 1000 is 3, or log10 (1000) = 3. The logarithm of x to base b is denoted as logb (x), or without parentheses, logb x, or even without the explicit base, log x, when no confusion is possible, or when the base does not matter such as in big O notation.

Logarithmic scale

A logarithmic scale (or log scale) is a way of displaying numerical data over a very wide range of values in a compact way. As opposed to a linear number line in which every unit of distance corresponds to adding by the same amount, on a logarithmic scale, every unit of length corresponds to multiplying the previous value by the same amount. Hence, such a scale is nonlinear: the numbers 1, 2, 3, 4, 5, and so on, are not equally spaced. Rather, the numbers 10, 100, 1000, 10000, and 100000 would be equally spaced.

Exponential function

The exponential function is a mathematical function denoted by or (where the argument x is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, although it can be extended to the complex numbers or generalized to other mathematical objects like matrices or Lie algebras. The exponential function originated from the notion of exponentiation (repeated multiplication), but modern definitions (there are several equivalent characterizations) allow it to be rigorously extended to all real arguments, including irrational numbers.

Logarithmic mean

In mathematics, the logarithmic mean is a function of two non-negative numbers which is equal to their difference divided by the logarithm of their quotient. This calculation is applicable in engineering problems involving heat and mass transfer. The logarithmic mean is defined as: for the positive numbers x, y. The logarithmic mean of two numbers is smaller than the arithmetic mean and the generalized mean with exponent one-third but larger than the geometric mean, unless the numbers are the same, in which case all three means are equal to the numbers.

Trigonometric functions

In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.

Related lectures (64)

Logarithmic Functions: Properties and Applications MOOC: Trigonometric, Logarithmic and Exponential Functions

Explores the properties and applications of natural logarithmic functions, including area invariance and harmonic series divergence.

Exponential Functions and Logarithms MOOC: Trigonometric, Logarithmic and Exponential Functions

Explores exponential and logarithmic functions, their properties, relations, and graphs, including different bases and commonly used logarithms.

Exponential Functions: Important Examples MOOC: Trigonometric, Logarithmic and Exponential Functions

Explores important examples of exponential functions and their limits, including lim (1 + ²)² = 0 and lim ex as x→∞.

Exponential Functions: Properties and Logarithms

Covers the properties of exponential and logarithmic functions.

Trigonometric Formulas MOOC: Trigonometric, Logarithmic and Exponential Functions