অভিসারী (কনভারজেন্ট) ধারা

####Background

If you need a primer on series in general, check out this explanation. It gives an introduction to what a series is and dips a bit into the ideas of convergent and divergent series, so it's a good place to start. In these lessons, you will cover sum of an infinite series and sigma notation (finding the sum). It might even be helpful to review our lessons on repeating decimals and rational numbers.

A convergent series is a series whose partial sums tend to a specific number, also called a limit. A divergent series is a series whose partial sums, by contrast, don't approach a limit. Divergent series typically go to $\infty$, go to $-\infty$, or don't approach one specific number.

An easy example of a convergent series is $$\sum\limits_{n=1}^{\infty}\frac{1}{2^n} = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots$$ The partial sums look like $$\frac{1}{2}, \frac{3}{4}, \frac{7}{8}, \frac{15}{16}, \cdots$$ and we can see that they get closer and closer to $1$. The first partial sum is $\frac{1}{2}$ away, the second $\frac{1}{4}$ away, and so on and so forth until it is infinitely close to $1$. This means that the above series converges to a limit at $1$!

(The ! does not indicate a factorial, although in the case of $1! = 1$ it wouldn't matter. Limits are just exciting.)

The convergence of different series is a very relevant topic in higher math and many significant mathematical discoveries revolve around different series, but for the purposes of Algebra 2 we can mainly concern ourselves with whether or not geometric series will converge.

Given a geometric series with common ratio $r$, it will converge whenever $|r| < 1$. This should make intuitive sense because if we had a geometric series with a common ratio of say, $2$, we would have something that looks like $$\sum\limits_{n=0}^{\infty}2^n = 1 + 2 + 4 + 8 + 16 + \cdots$$ Clearly the numbers we are adding to the sum get larger and larger, so the sum would never slow down and approach a specific limit. This is an example of a divergent series.

Similarly, if the common ratio is $-2$, the series $$\sum\limits_{n=0}^{\infty}(-2)^n = 1 - 2 + 4 - 8 + 16 - 32 + \cdots$$ will also diverge. A way to better see this is $$\begin{align} \sum\limits_{n=0}^{\infty}(-2)^n &= (1-2)+(4-8)+(16-32)+ \cdots \\ &= -1 - 4 - 16 - \cdots \\ &= -\sum\limits_{n=0}^{\infty}4^n \end{align}$$ which is clearly divergent.

Now that you've been introduced to convergence and divergence, feel free to try out some of the problems in this topic. Have fun!

Infinite Geometric Series as Rational Functions

An infinite geometric series is a geometric series that goes on forever. Even though the series never ends, it can sometimes be written as one simple fraction. We've learned about this type of fraction before: the rational function. We call it that because it is made from polynomials.

Before applying other convergence tests to a geometric series, you can use the Nth Term Test for Divergence of Series.

For example,

$$1 + x + x^2 + x^3 + ...$$

can be written as:

$$\frac{1}{1-x}$$

These two expressions look very different, but they represent the same value whenever $|x|<1$. Writing an infinite series as a rational function makes it much easier to work with. Instead of adding infinitely many terms, you can use one simple expression.

This connection between infinite geometric series and rational functions is useful throughout algebra, calculus, and many real-world applications.

Like Finite Telescoping Sums and Products, Infinite Geometric Series as Rational Functions show how a complicated-looking series can simplify to a compact algebraic expression. Both topics emphasize recognizing patterns that lead to closed-form results.

Sum of Infinite Geometric Series

by Mario's Math Tutoring

In this lesson by Mario's Math Tutoring, you will learn how to find the sum of an infinite geometric series.