Does the Infinite Series Converge or Diverge? A Problem on Complex Numbers

In summary, the conversation discusses the determination of convergence or divergence of an infinite series and the use of the ratio test to solve it. The individual asking for help applied the ratio test and attempted to simplify the expression further, but ultimately needed additional advice. Another individual suggests using a comparison of absolute values instead of the ratio test.
  • #1
matpo39
43
0
i need a little help with this problem:

determine if the infinite series converges or diverges.
summation (from n=1..infinity) {1/(n^2+i^n)}

I first applied the ration test to this series and got

(n+1)^2 + i^(n+1) / [n^2 + i^n]

i then multiplied top and bottom by (n^2 - i^n)
which gave

[{(n+1)^2 + i^(n+1) }* (n^2-i^n) ]/{n^4 - i^2n}

this is where i get stuck, i can't seem to simplify it any further.
if some one can give me some advice on it, it would be greatly appreciated.

thanks
 
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  • #2
the ratio test is the best way to do it. but when you've got an expression like

[tex]\frac{(n+1)^2+c}{n^2+d}[/tex]

all you need to do is divide every term by n^2.
 
  • #3
Also don't forget that it is the limit of the absolute value of the ratio that counts.
 
  • #4
I don't think the ratio test works here. A simple comparison (of the terms' absolute values) seems to work better.
 
Last edited:

1. What is a complex infinite series?

A complex infinite series is a mathematical expression that involves an infinite number of terms, where each term is a complex number. It is written in the form of Σ cn, where n ranges from 0 to infinity and cn is a complex number.

2. How is a complex infinite series different from a regular infinite series?

A regular infinite series involves only real numbers, while a complex infinite series involves complex numbers, which can have both real and imaginary parts. This adds an extra layer of complexity to the series and requires different methods for analysis and convergence.

3. What is the importance of complex infinite series?

Complex infinite series are important in many areas of mathematics and physics. They are used to represent functions, solve differential equations, and approximate real-world phenomena. They also have applications in engineering, signal processing, and quantum mechanics.

4. How do you determine the convergence of a complex infinite series?

The convergence of a complex infinite series is determined by analyzing the behavior of the series' terms. The most commonly used tests for convergence are the ratio test, root test, and comparison test. These tests check the behavior of the series as n approaches infinity to determine if it converges or diverges.

5. Are there any practical applications of complex infinite series?

Yes, complex infinite series have many practical applications in fields such as engineering, physics, and economics. For example, they are used to model electrical circuits, analyze heat transfer, and predict stock market trends. They also play a crucial role in signal processing, where they are used to filter and analyze data in various forms.

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