What is the ratio test for proving absolute convergence of a series?

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In summary, The conversation revolves around proving the absolute convergence of a power series for all values of w within its radius of convergence. The suggestion is to use the squeeze theorem and the ratio test to prove this. The idea of having different coefficients for each term in the power series is also discussed.
  • #1
steven187
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hello all

well i think I am kind of brain dead, iv been workin on a lot of problems over the last few days, I can't see anything obvious anymore, well this shall be the last one for today (i hope), anyway here it is,

suppose that for some [tex]x\not= 0 [/tex], the series
[tex]\sum_{n=1}^{\infty} a_n x^n[/tex]
is convergent. Prove the series is absolutely convergent for all [tex]w[/tex] with [tex]|w|<|x|[/tex].

Steven
 
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  • #2
Well, a power series diverges outside its radius of convergence and converges absolutely on the inside...
 
  • #3
Try the squeeze theorem. What's with a_n, though? You mean each term has a different coefficient?
 
  • #4
Icebreaker said:
Try the squeeze theorem. What's with a_n, though? You mean each term has a different coefficient?

Well, yes! That is the basic idea of a power series after all.
 
  • #5
Odd, I have the idea in my head that they must have the same coefficient in order to find its sum, if it's convergent.
 
  • #6
Hello Steven.

How about using the ratio test:

If:

[tex]\mathop\lim\limits_{n\to\infty} |\frac{u_{n+1}}{u_n}|=L<1[/tex]

then the given series is absolutely convergent.
 

What is the definition of absolutely convergent?

Absolute convergence refers to a mathematical series in which the sum of the absolute values of the terms converges to a finite value. This means that even if the individual terms in the series may alternate in sign and possibly cancel each other out, the overall sum still converges.

How is absolute convergence different from conditional convergence?

Conditional convergence is a type of convergence in which the sum of a series depends on the order in which the terms are added. In other words, rearranging the terms of a conditionally convergent series can lead to a different sum. Absolute convergence, on the other hand, is not affected by the order of the terms and will always converge to the same value.

What is the significance of absolute convergence in mathematics and science?

Absolute convergence is important in mathematics because it allows for easier calculations and simplification of complex series. In science, it is often used in the analysis of data and in the development of mathematical models to accurately predict outcomes.

How can you determine if a series is absolutely convergent?

To determine absolute convergence, you can use several tests such as the ratio test, the comparison test, or the root test. These tests evaluate the behavior of the terms in a series and can determine if the series converges absolutely, conditionally, or diverges.

What are some real-world applications of absolute convergence?

Absolute convergence has many applications in fields such as physics, engineering, and economics. It is used to analyze data, develop mathematical models, and make predictions about future outcomes. In physics, it is used to study the behavior of electric and magnetic fields, while in economics, it is used to analyze financial data and make predictions about market trends.

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