Geometric sequences graphic representations

One intuitive example of how to sum a geometric series. A geometric series of ratio less than 1 is convergent.

The geometric series of ratio 1/2 is convergent. We can represent this series using a rectangle and cut it in half successively. Here we use a rectangle such us all rectangles are similar.

Gamma, the Euler's constant, is defined using a covergent series.

Power with natural exponents are simple and important functions.

We extend the definition of power of positive integer exponent and their inversas if we consider positive rational exponents.

We can see how the graph of a polynomial of degree three with three real roots changes when we change a root.

The powers of natural exponent are the base of the polynomials. We can consider the polynomic function that passes through a series of points of the plane. This is an interpolation problem that is solved here using the Lagrange interpolating polynomial.

The integral concept is associate to the concept of area. We began considering the area limited by the graph of a function and the x-axis between two vertical lines.

When approximating an integral using rectangles we commit an error. In some cases, for example, in monotonic functions, we can limit the magnitude of the error.

Archimedes show us in 'The Method' how to use the lever law to discover the area of a parabolic segment.

Studying the volume of a barrel, Kepler solved a problem about maxima in 1615.

By increasing the degree, Taylor polynomial approximates the exponential function more and more.

Mercator published his famous series for the Logarithm Function in 1668. Euler discovered a practical series to calculate.

By increasing the degree, Taylor polynomial approximates the sine function more and more.

The function is not defined for values less than -1. Taylor polynomials about the origin approximates the function between -1 and 1.

The function has a singularity at -1. Taylor polynomials about the origin approximates the function between -1 and 1.

The function has a singularity at -1. Taylor polynomials about the origin approximates the function between -1 and 1.

This function has two real singularities at -1 and 1. Taylor polynomials approximate the function in an interval centered at the center of the series. Its radius is the distance to the nearest singularity.

This is a continuos function and has no real singularities. However, the Taylor series approximates the function only in an interval. To understand this behavior we should consider a complex function.

Mercator published his famous series for the Logarithm Function in 1668. Euler discovered a practical series to calculate.