Infinite series with all negative terms

In summary, an infinite series with all negative terms is a mathematical series where each term has a negative value and continues indefinitely. The main difference between a finite and an infinite series with all negative terms is that the former has a fixed number of terms while the latter has an infinite number of terms. The sum of an infinite series with all negative terms can be calculated using a specific formula, and it can have a finite sum if the common ratio between terms is less than 1. This type of series has various real-life applications in fields such as economics, physics, and engineering.
  • #1
Ibraheem
51
2
Hello,

I have been reviewing my textbook lately, and I came across a rather paradoxical statements. all of the convergence tests in my book state that the terms of the series has to be positive. However, when I solved this power series ∑(-1)n-1(xn/n), I found that it converges for -1<x≤1, but when I plugged -0.45 to test if the infinite series will converge, I got all the terms negative. So my question is can a series converge if all its terms are negative?
 
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  • #2
Yes. E.g. ##-e##.
 

What is an infinite series with all negative terms?

An infinite series with all negative terms is a mathematical series where each term has a negative value. It can be written in the form of a sum, where the terms continue indefinitely.

What is the difference between a finite and an infinite series with all negative terms?

A finite series with all negative terms has a fixed number of terms, while an infinite series with all negative terms has an infinite number of terms. In other words, a finite series has an end point, while an infinite series does not.

How is the sum of an infinite series with all negative terms calculated?

The sum of an infinite series with all negative terms can be calculated using the formula Sn = a/(1-r), where Sn is the sum, a is the first term, and r is the common ratio between terms. This formula only applies if the absolute value of r is less than 1.

Can an infinite series with all negative terms have a finite sum?

Yes, an infinite series with all negative terms can have a finite sum if the absolute value of the common ratio between terms is less than 1. In this case, the series will converge to a specific value as the number of terms approaches infinity.

What are some real-life applications of infinite series with all negative terms?

Infinite series with all negative terms can be used in various fields, such as economics, physics, and engineering. For example, they can be used to model the behavior of radioactive decay, population growth, or the value of investments over time.

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