Infinite Series: A Beginner's Guide

In summary, infinite series is a mathematical concept where the sum of an infinite number of terms is calculated. It can be thought of as taking steps of halving distance, with each step getting smaller and smaller. The sum of these steps can approach a finite value, as seen in the example of 1 + 1/2 + 1/4 + 1/8 +1/16 + etc. This concept is used in solving various homework problems and can be calculated using the formula T_n = ar^{n-1}, where r < 1 and S_\infty = \frac{a}{1-r} as n approaches infinity and r^n approaches 0.
  • #1
vigintitres
26
0
I am getting into this topic and I am having a hard time conceptualizing it. Is there anybody that can spend a minute letting me know the "reality" to infinite series? By that I mean, please explain infinite series in such a way that a beginner like me will be able to take what you said and apply it to home work problems. Thanks and if I am asking too much I understand because this is a deep topic
 
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  • #2
I don't know what you mean by reality. However a simple example is:

1 + 1/2 + 1/4 + 1/8 +1/16 + etc. where you never stop. As a result it adds up to 2.
 
  • #3
mathman said:
I don't know what you mean by reality. However a simple example is:

1 + 1/2 + 1/4 + 1/8 +1/16 + etc. where you never stop. As a result it adds up to 2.

You can think of this as taking half a step as you did last step. Your first step is of 1 unit, so next step is 1/2, next one 1/4 and so on. The question is how much distance can you travel using this method and that is 2. You cannot exceed 2 units of distance by using this mode of transportation.
 
  • #4
[tex]
T_n = ar^{n-1}
[/tex]

[tex]
r < 1
[/tex]

[tex]
S_n = \frac{a(1-r^n)}{1-r}
[/tex]

[tex]
n \rightarrow \infty
[/tex]

[tex]
r^n \rightarrow 0
[/tex]

[tex]
S_\infty = \frac{a}{1-r}
[/tex]
 

1. What is an infinite series?

An infinite series is a mathematical concept that involves adding an infinite number of terms together in a specific order. It is represented in the form of ∑(n=1 to ∞)an, where an is the nth term of the series.

2. How is an infinite series different from a finite series?

An infinite series has an infinite number of terms, while a finite series has a limited number of terms. This means that an infinite series does not have a final sum, whereas a finite series has a finite sum.

3. What is the difference between a convergent and a divergent infinite series?

A convergent infinite series is one in which the sum of all the terms approaches a finite number as the number of terms increases. In contrast, a divergent infinite series is one in which the sum of the terms does not approach a finite number, but either increases or decreases without bound.

4. How do you determine if an infinite series is convergent or divergent?

There are various tests that can be used to determine the convergence or divergence of an infinite series, such as the comparison test, the ratio test, and the integral test. These tests involve comparing the series to a known convergent or divergent series or using calculus techniques to analyze the behavior of the series.

5. What are some real-world applications of infinite series?

Infinite series have many practical applications in fields such as physics, finance, and engineering. They are used to model natural phenomena, calculate probabilities, and solve optimization problems. For example, infinite series are used in the calculation of compound interest in finance and in the study of motion and forces in physics.

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