Test for convergence of a series

Thread starter Michael_Light
Start date Feb 5, 2012
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Convergence Series Test

In summary, a series is a mathematical concept of adding together a sequence of numbers. It is related to convergence, which refers to whether the sum of the series approaches a finite value or diverges to infinity. To determine convergence, various tests such as the comparison, ratio, and integral tests can be used. Absolute convergence means the series converges regardless of the order of terms, while conditional convergence only converges in a specified order. A series can only converge to one value, and the rate of convergence affects how quickly it approaches its limit, potentially indicating stability and predictability.

Feb 5, 2012

Michael_Light

Homework Statement

Homework Equations

The Attempt at a Solution

I have no ideas how to continue. I also tried the comparison test but I don't know where to start. Please guide me...

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Feb 5, 2012

Dick

Science Advisor

Homework Helper

You want the limit of the ratio as n->infinity. Look at just the [itex]\frac{(n+1)^k}{n^k}[/itex] part. That's the same as [itex](\frac{n+1}{n})^k=(1+\frac{1}{n})^k[/itex]. What is that limit?

Related to Test for convergence of a series

1. What is a series and how is it related to convergence?

A series is a mathematical concept in which a sequence of numbers is added together. It is related to convergence in that the convergence of a series refers to whether the sum of the series approaches a finite value as the number of terms increases, or if it diverges and approaches infinity.

2. How do you determine if a series converges or diverges?

To determine if a series converges or diverges, you can use various tests such as the comparison test, ratio test, or integral test. These tests examine the behavior of the terms in the series and determine if the series approaches a finite limit or not.

3. What is the difference between absolute and conditional convergence?

Absolute convergence refers to a series that converges regardless of the order in which the terms are added. On the other hand, conditional convergence refers to a series that only converges when the terms are added in a specific order. This means that rearranging the terms of a conditionally convergent series can change its sum.

4. Can a series converge to more than one value?

No, a series can only converge to one value. If a series converges, it means that the sum of the terms approaches a specific finite value. If the series were to converge to multiple values, then the sum would not have a definite value.

5. How does the rate of convergence affect the behavior of a series?

The rate of convergence determines how quickly a series approaches its limit. A series with a faster rate of convergence will approach its limit more quickly than a series with a slower rate of convergence. This can affect the behavior of the series, as a faster rate of convergence may indicate a more stable and predictable series.