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Convergent definition in mathematics is a property (displayed by certain innumerable series and functions) of approaching a limit more and more explicitly as an argument (variable) of the function increases or decreases or as the number of terms of the series gets increased. For instance, the function y = 1/x converges to zero (0) as increases the 'x'. Even so, no finite value of x will influence the value of y to really become zero, the limiting value of y is zero (0) since y can be made as small as wanted by selecting 'x' huge enough. The line y = 0 (the x-axis) is known as an asymptote of the function.

In the same manner as the above example, for any value of x between (but exclusive of) +1 and -1, the series 1 + x + x^{2} + ⋯ + x^{n} converges toward the limit 1/(1 − x) as n, the number of terms, increases. The interval −1 < x < 1 is known as the range of convergence of the series; for values of x on the exterior of this range, the series is declared to diverge.

Convergence usually means coming together, whereas divergence usually implies moving apart. In the world of trade and finance, convergence and divergence are terms used to define the directional association of two prices, trends or indicators.

A convergent sequence, a sequence of numbers in which number come ever near from a real number (known as the limit):

For ex; 70, 80, 90, 95, 97, 98, 99, 99.5, 99.8, 99.9, 99.999….

Looking at this sequence, you are most likely to surmise that the numbers always come closer to 100, and you’d be right.

Other examples of convergent sequences include:

0, 1, 2, 2, 2, 2, 2, 2, 2, 2…

The rule here is: keep adding +1 to the preceding number until you reach 2, then put a pause. The limit is thus 2.

64, 32, 16, 8, 4, 2, 1, 0.5, 0.25, 0.125...

Here every number is just half of the previous one. The limit is ZERO (0). (No term of the sequence will ever reach up to zero; it’ll just keep coming infinitely closer from it)

Now a divergent sequence, any sequence that does NOT come closer from a real number.

Either because its limit is infinite:

For example:

2, 4, 8, 16, 32, 64, 128, 256, and 512 1024, 2048, 4096…

In this sequence, every number is double the preceding number (U (n+1) = 2*U_{n}). It will keep increasing, infinitely. Because its limit, infinity, is NOT a real number, is said to be a sequence infinite.

You must have understood the convergent math definition, now let's proceed to solve the numerical problem associated with the concept.

Example: Evaluate if the given series converges or diverges. If it converges find out its sum.

\[\int_{n=1}^{\infty} (\frac{1}{3})^{n-1}\]

Solution:

Using the general formula for the partial sums for this series i.e.

\[s_{n} = \int_{i=1}^{n} (\frac{1}{3})^{i - 1} = \frac{3}{2} (1 - 1/3)^{n}\]

In the given example, the limit of the sequence of partial sums is,

\[\lim_{n \rightarrow \infty} s_{n} = \lim_{n \rightarrow \infty} \frac{3}{2} (1 - 1/3)^{n} = \frac{3}{2}\]

We come to the conclusion that the sequence of partial sums is convergent and thus the series will also be convergent and the value of the series is as follows:

\[\int_{n=1}^{\infty} (\frac{1}{3})^{n-1} = \frac{3}{2}\]

Example:

Evaluate if the given series converges or diverges. If it converges find out its sum.

\[\int_{n=2}^{\infty} \frac{1}{n^2} - 1\]

Solution:

This is one of the few series where we are able to identify a formula for the general term in the sequence of partial fractions.

The general formula used for the partial sums is as below;

\[S_{n} = \int_{n=2}^{\infty} \frac{1}{i^{2}} - 1 = 3/4 - 1/2n - 1/2(n+1)\]

And in this example, we have,

\[\lim_{n \rightarrow \infty} s_{n} = \lim_{n \rightarrow \infty} (3/4 - 1/2n - 1/2 (n+1)) = 3/4\]

The sequence of partial sums converges and thus the series converges. The value of the series is,

\[\int_{n=2}^{\infty} \frac{1}{n^2} - 1 = 3/4\]

FAQ (Frequently Asked Questions)

Q1. What is Uniform Convergence?

Answer: Uniform convergence, in mathematical analysis is a property involving the process of convergence of an order of continuous functions —f_{1}(x), f_{2}(x), f_{3}(x), f_{4}(x), f_{5}(x), f_{6}(x), f_{7}(x)…to a function f(x) for all x in some interval (a, b). Specifically, for any positive number ε > 0 there remain a positive integer N for which |f_{n}(x) − f(x)| ≤ ε for all n ≥ N and all x in (a, b). In point-by-point convergence, N depends on both the closeness of ε and the specific point x.

Q2. What is Divergence in Math?

Answer: Divergence, in mathematical terms, is a differential operator applied to a 3D vector-valued function. The outcome is typically a function which defines a rate of change. The divergence of a vector v is provided by divergence of a vector "v" where v_{1}, v_{2}, and v_{3}, v_{4} are the vector components of v, essentially a velocity field of fluid flow.

Q3. What is Divergence in Math?

Answer: In mathematics, power series is an infinite series that can be considered as a polynomial having an infinite number of terms, in a way that 1 + x + x^{2} + x^{3} + x^{4} + x^{5}+ x^{6} + x^{7}… In general, a given power series is supposed to converge (i.e., approach a finite sum) for all values of x within a particular interval around zero (0) - specifically, whenever the absolute value of x is less than some positive number r, called as the radius of convergence. The series outside of this interval diverges (is infinite), whereas the series may converge or diverge if x = ± r. The radius of convergence can frequently be evaluated by a version of the ratio test for power series: given a general power series a_{0} + a_{1}x + a_{2}x^{2} +⋯, where the coefficients are known.