matematicas visuales visual math

If we start with a decreasing and positive function

Convergence of Series, Integral Test | matematicasVisuales

You can define

Convergence of Series, Integral Test | matematicasVisuales

And then you get a series of positive terms

Convergence of Series, Integral Test | matematicasVisuales
Convergence of Series, Integral Test: a positive decreasing function and a series | matematicasVisuales

Convergence of Series, Integral Test: the integral of the function | matematicasVisuales
Convergence of Series, Integral Test: series as a sum or rectangles | matematicasVisuales

In general, you can say that this inequality holds:

Convergence of Series, Integral Test | matematicasVisuales

In the mathlet you can play with a particular case

Convergence of Series, Integral Test | matematicasVisuales

Dragging the dots you can change the values of lambda and p.

In theses cases, the series that you get is like a p-series (translated and expanded).

Some of these integrals diverge then the series diverges too:

Convergence of Series, Integral Test | matematicasVisuales
Convergence of Series, Integral Test: some series diverge | matematicasVisuales

This is the case when in a p-series p es equal or greater than 1. The integral and the series diverge if you 'crosses this line' dragging the green dots:

Convergence of Series, Integral Test | matematicasVisuales

In some other cases, the integral and the series converges:

Convergence of Series, Integral Test: a series that converges, lower and upper bound | matematicasVisuales

In the mathlet, click the play button to see the animation. You can see that the series is the integral plus something that is less than ak.

And we can say that

Convergence of Series, Integral Test | matematicasVisuales

Better than that, you get a lower and an upper bounds of the series.

Convergence of Series, Integral Test | matematicasVisuales

For example, consider the integral

Convergence of Series, Integral Test: a series that converges, example and lower and upper bounds | matematicasVisuales
Convergence of Series, Integral Test: a series that converges, example and lower and upper bounds| matematicasVisuales

Then

The series converges and the lower and upper bounds are

REFERENCES

These are classical results but this page is directly inspired in Jim Fowler´s lesson 'How can integrating help us to address convergence'. This is part of the Coursera course Calculus Two: Sequences and Series that Jim Fowler is teaching with all his enthusiasm (October 2013).

MORE LINKS

Sum of a geometric series of ratio 1/4
One intuitive example of how to sum a geometric series. A geometric series of ratio less than 1 is convergent.
Sum of a geometric series of ratio 1/2
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.
Definite integral
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.
Indefinite integral
If we consider the lower limit of integration a as fixed and if we can calculate the integral for different values of the upper limit of integration b then we can define a new function: an indefinite integral of f.
Polynomial functions and integral (2): Quadratic functions
To calculate the area under a parabola is more difficult than to calculate the area under a linear function. We show how to approximate this area using rectangles and that the integral function of a polynomial of degree 2 is a polynomial of degree 3.
Polynomial functions and integral (3): Lagrange polynomials (General polynomial functions)
We can see some basic concepts about integration applied to a general polynomial function. Integral functions of polynomial functions are polynomial functions with one degree more than the original function.
The Fundamental Theorem of Calculus (1)
The Fundamental Theorem of Calculus tell us that every continuous function has an antiderivative and shows how to construct one using the integral.
The Fundamental Theorem of Calculus (2)
The Second Fundamental Theorem of Calculus is a powerful tool for evaluating definite integral (if we know an antiderivative of the function).
Taylor polynomials (1): Exponential function
By increasing the degree, Taylor polynomial approximates the exponential function more and more.
Taylor polynomials (2): Sine function
By increasing the degree, Taylor polynomial approximates the sine function more and more.
Taylor polynomials (3): Square root
The function is not defined for values less than -1. Taylor polynomials about the origin approximates the function between -1 and 1.
Taylor polynomials (7): Rational function without real singularities
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.
Complex Polynomial Functions(2): Polynomial of degree 2
A polynomial of degree 2 has two zeros or roots. In this representation you can see Cassini ovals and a lemniscate.
Complex Polynomial Functions(3): Polynomial of degree 3
A complex polinomial of degree 3 has three roots or zeros.
Complex Polynomial Functions(4): Polynomial of degree n
Every complex polynomial of degree n has n zeros or roots.
The Complex Exponential Function
The Complex Exponential Function extends the Real Exponential Function to the complex plane.
The Complex Cosine Function
The Complex Cosine Function extends the Real Cosine Function to the complex plane. It is a periodic function that shares several properties with his real ancestor.
The Complex Cosine Function: mapping an horizontal line
The Complex Cosine Function maps horizontal lines to confocal ellipses.
Inversion
Inversion is a plane transformation that transform straight lines and circles in straight lines and circles.
Multifunctions: Powers with fractional exponent
The usual definition of a function is restrictive. We may broaden the definition of a function to allow f(z) to have many differente values for a single value of z. In this case f is called a many-valued function or a multifunction.
Multifunctions: Two branch points
Multifunctions can have more than one branch point. In this page we can see a two-valued multifunction with two branch points.
Taylor polynomials: Complex Exponential Function
The complex exponential function is periodic. His power series converges everywhere in the complex plane.
Taylor polynomials: Complex Cosine Function
The power series of the Cosine Function converges everywhere in the complex plane.
Taylor polynomials: Rational function with two complex singularities
We will see how Taylor polynomials approximate the function inside its circle of convergence.