Does the Series Converge or Diverge?

In summary, a divergent series is a series where the terms do not approach a finite limit, an absolutely convergent series is a series where the absolute values of the terms converge to a finite limit, and a conditionally convergent series is a series where the terms themselves converge to a finite limit but the series does not converge if the absolute values of the terms are taken. To determine if a series is divergent, one can look at the behavior of the terms or use tests such as the divergence test or the integral test. Absolute convergence is important as it guarantees convergence to a finite value regardless of the order of terms, and it can be determined by tests such as the ratio test, root test, or comparison test. Real-life applications of
  • #1
shamieh
539
0
determine if series is absolutely convergent, conditionally convergent, or divergent

\(\displaystyle \sum^{\infty}_{n = 1} (n^2 + 9)(-2)^{1-n} \)

which i turned into \(\displaystyle \sum^{\infty}_{n = 1} (n^2 + 9)(-2)^{-n+1} \)

so using the ratio test I got:

\(\displaystyle \frac{((n+1)^2 + 9)(-2)^{-n})}{n^2 + 9 * (-2)^{1-n}}\)

which ended up as n--> infinity becoming \(\displaystyle \frac{2}{3}\) therefore by ratio test L < 1 so the series converges
 
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  • #2
nvm I got 1/2
 

FAQ: Does the Series Converge or Diverge?

1. What is the difference between a divergent series, an absolutely convergent series, and a conditionally convergent series?

A divergent series is a series where the terms do not approach a finite limit as the number of terms increases. An absolutely convergent series is a series where the absolute values of the terms converge to a finite limit. A conditionally convergent series is a series where the terms themselves converge to a finite limit, but the series does not converge if the absolute values of the terms are taken.

2. How do you determine if a series is divergent?

A series is divergent if the terms do not approach a finite limit as the number of terms increases. This can be determined by looking at the behavior of the terms in the series, such as whether they increase or decrease without approaching a limit, or by using tests such as the divergence test or the integral test.

3. What is the significance of absolute convergence in series?

Absolute convergence is important because it guarantees that the series will converge to a finite value regardless of the order in which the terms are added. This allows for easier manipulation and calculation of series, as well as making it easier to determine whether a series is convergent or not.

4. How can you determine if a series is absolutely convergent?

A series is absolutely convergent if the absolute values of the terms converge to a finite limit. This can be determined by using tests such as the ratio test, root test, or comparison test. If the series passes one of these tests and the absolute values of the terms converge, then the series is absolutely convergent.

5. What are some real-life applications of determining if a series is conditionally convergent or absolutely convergent?

One real-life application is in finance, where absolute convergence is used to calculate the present value of a future cash flow. In physics, absolute convergence is used to calculate the total energy of a system. In engineering, absolute convergence is used to determine the convergence of numerical methods used for solving complex equations. In general, determining the convergence of series can help in making accurate calculations and predictions in various fields.

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