Convergence/Divergence of an infinite series

In summary, To prove the convergence or divergence of the series \Sigma(\frac{n}{2n+3})^{2} using the Direct Comparison Test, we can simplify the series by dividing both the numerator and denominator by n. This yields \frac{1}{4+\frac{3}{n}} which approaches 0 as n approaches infinity. However, this does not provide a useful comparison for the Direct Comparison Test. Therefore, we cannot determine the convergence or divergence of the given series using this method.
  • #1
rdioface
11
0

Homework Statement


Prove the convergence or divergence of the series [tex]\Sigma[/tex][tex](\frac{n}{2n+3})^{2}[/tex] using the Direct Comparison Test.


Homework Equations


If series A converges and every term in series B is less than the corresponding term in series A, then series B converges.
If series C diverges and every term in series D is greater than the corresponding term in series C, then series D diverges.


The Attempt at a Solution


I can't think of anything useful to compare it to. Turning either the top or the bottom to 1 doesn't yield anything useful, nor does it behave as a p-series.
 
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  • #2
hi rdioface! :smile:

(use "left(" and "right)" to automatically get latex brackets the right size :wink:)

always try to simplify by getting things as close to 1 as possible …

in this case, try divdiding top and bottom by n :smile:
 

Related to Convergence/Divergence of an infinite series

1. What is the definition of convergence of an infinite series?

The convergence of an infinite series is a mathematical concept that refers to the behavior of a series as the number of terms in the series increases to infinity. A series is said to converge if the sum of its terms approaches a finite value, known as the limit, as the number of terms increases.

2. How can I determine if an infinite series converges or diverges?

There are several tests that can be used to determine the convergence or divergence of an infinite series, including the comparison test, the ratio test, and the integral test. These tests involve comparing the series to other known series or using calculus techniques to analyze the behavior of the series.

3. What is the difference between absolute and conditional convergence of an infinite series?

Absolute convergence refers to a series in which the sum of the absolute values of the terms converges. Conditional convergence refers to a series in which the sum of the terms converges, but the sum of the absolute values of the terms diverges. In other words, conditional convergence is a weaker form of convergence than absolute convergence.

4. How can I use the concept of convergence to evaluate the sum of an infinite series?

If a series is known to converge, the sum of the series can be calculated by taking the limit of the partial sums of the series as the number of terms approaches infinity. This is known as the sum of an infinite series or the limit of a sequence.

5. Can an infinite series converge to more than one value?

No, an infinite series can only converge to one value. If the series converges, the limit of the partial sums will approach a single finite value. If the series does not converge, there is no single value that the partial sums approach, and the series is said to diverge.

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