Convergence in Mathematics

Dhristi JEE 2022-24

Convergent Meaning in Maths

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 it 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.


Divergent Convergent Math

In the same manner as the above example, for any value of x between (but exclusive of) +1 and -1, the series 1 + x + x2 + ⋯ + xn converges towards 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.


Difference Between Convergent and Divergent Math

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 numbers come ever near from a real number (known as the limit):

For example, 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*Un). It will keep increasing, infinitely. Because its limit, infinity, is NOT a real number, it is said to be a sequence infinite.


Solved Examples

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.

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,

\[s_{n} = \int_{i=0}^{n} (\frac{1}{3})^{i-1} = \frac{3}{2} (1-1/3)^{n}\]

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.

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

FAQs on Convergence in Mathematics

1. What is uniform convergence?

Uniform convergence, in mathematical analysis, is a property involving the process of convergence of an order of continuous functions —f1(x), f2(x), f3(x), f4(x), f5(x), f6(x), f7(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 |fn(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.

2. What is divergence in Math?

Divergence, in mathematical terms, is a differential operator applied to a 3D vector-valued function. The outcome is typically a function that defines a rate of change. The divergence of a vector v is provided by the divergence of a vector "v" where v1, v2, and v3, v4 are the vector components of v, essentially a velocity field of fluid flow.

3. What is an infinite series in Math?

In mathematics, power series is an infinite series that can be considered as a polynomial h3. What is an infinite series in Math?aving an infinite number of terms, in a way that 1 + x + x2 + x3 + x4 + x5+ x6 + x7… 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 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 a0 + a1x + a2x2 +⋯, where the coefficients are known.

4. What is the difference between convergent and divergent series?

Number sequences are divided into finite sequence and infinite sequence. The infinite sequence is further divided into convergent and divergent series. The major difference between convergent and divergent series is the limit, the former has a limit and the latter series does not have any limit. Divergent series are further divided into ending with infinity or negative infinity which is far away from reaching zeros. 1, ½,⅓,¼,⅕… is an example of convergent series as it approaches “0” when extended but the series -1,-2,-3,-4,… do not end with a limit, and hence, they can be considered a divergent series that explodes in negative infinity. On the other hand, there is another type of divergent series like 1,1,2,3,4,1,3,2,4…. In this series, the numbers just move front and back and reach infinite hence, this type of series is also called divergent series. To know more about divergent and convergent series comprehensively, read the above article from Vedantu. 

5. Give some examples of convergent and divergent series.

Number series are basically of two types, namely finite and infinite sequences. This division is further extended to convergent and divergent series. The major difference between divergent and convergent series is the series reaching a limit. Convergent series ends up with a limit, hence it is a finite series and divergent series do not reach a real number as limit and can be extended infinitely. Following are some of the examples of convergent and divergent series:

  • When the series, 1, ½, ⅓, ¼, ⅕,… is extended, it reaches “0” which is a real number at some point. Hence, this series is called the convergent series.

  • On the other hand, the series, -1, -2, -3, -4, -5, -6… is a divergent series as it moves into negative infinity away from zero and does not have a limit. 

  • Similarly, the series, 2, 1, 3, 4, 1, 2, 4, 2, 2, 2, 3, 4, 3, 4,… is also divergent as the same numbers repeat without a particular sequence and never reach the limit and are away from zero. This is another type of divergent series.

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