[Maths Class Notes] on Convergence in Mathematics Pdf for Exam

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]