Can any criterion be applied to calculate the convergence or divergence of an infinite series?

carlos84 (72)in Popular STEM • 8 days ago

A series is a succession, for this case are series that tend to infinity, so it is very conducive to know if these series converge or diverge.

For it we are going to apply a criterion called criterion of the integral, which consists of converting the series to the similar form of a function, if the function decreases in the interval of the series that goes from [1,∞)
then the integral criterion can be applied, which is nothing more than solving the integral in the interval of the series and whose integrand is the converted function of the series.

If the integral is convergent, then the series is convergent, while if the integral is divergent, then the series is divergent, everything said here is summarized in the following theorem:

Theorem 9.10: The integral criterion

If f is positive, continuous and decreasing for
y then it follows that:

Both the series and the integral converge or both diverge.

That is why if we conclude convergence or divergence for the integral, then it is the same result for the infinite series.

Application of the integral criterion

Apply the integral criterion to the series:

The first thing we are going to do is to convert it into its function-like form:

We realize that the function is positive, from the interval [1,∞) the function is continuous and is decreasing, as can be seen in the following graph generated with GeoGebra software:

As could be seen in the graph of the rational function it is fulfilled that in the interval [1,∞) the function is positive, continuous and decreasing, so the criterion of the integral can be applied as follows:

When analyzing the integral, we realize that it is an improper integral of the form:

Therefore, it is resolved as follows:

We solve the indefinite integral:

We do this by the method of variable substitution or change of variable, where u = x²+1

The change of variable is made and it remains:

The integral of du/u is equal to Ln(u), so it remains:

We return the change of variable:

We now evaluate the integral on the interval [1,∞) by applying the fundamental theorem of calculus:

Now we apply the limit when b has infinity, as follows:

Since the limit gives us infinity, that means that the series does not converge to any real value when the series grows to infinity, i.e. it keeps growing, so if the improper integral is divergent, then the series is divergent as well.