Determine whether the series converges or diverges.

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In summary, the conversation discusses the convergence of the series \sum cos(n*pi)/(n^(3/4)) and its representation as \sum((-1)^n)/(n)^(3/4). It is determined that the series converges due to it being an alternating series, and it cannot be tested using the p-series test. The idea of taking the absolute value of the series is also discussed, with the conclusion that it is not always true that the convergence of the absolute value series implies the convergence of the original series.
  • #1
Sabricd
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Hello,

I have to determine whether the given series diverges or converges. [tex]\sum[/tex] cos(n*pi)/(n^(3/4)) where n= 1 and goes to infinity.

I tried a couple numbers for n and got:

-1 + 1/(2)^(3/4) - 1/(3)^(3/4)

Hence I came up with the series: [tex]\sum[/tex]((-1)^n)/(n)^(3/4) where n=1 and goes to infinity.

I guess my main question is that now that I have that new representation of the series, why can't I just take the absolute value of the series and say that it is a p series with p< 1 and therefore diverges.

My book took the limit of the series and got 0 and said it converged. Why can't you use the p-series test?


Thank you,
 
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  • #3
So when you have alternating series you cannot use the p-series test?
 
  • #4
But the p-series is for series of the form

[tex]\sum_{n=1}^\infty{\frac{1}{n^p}}[/tex]

You have a factor [tex](-1)^n[/tex] extra.
I guess that you could say that you just took the absolute value of each term. But it is NOT TRUE to say that

[tex]\sum{x_n}~\text{diverges}~\text{if}~\sum{|x_n|}~\text{diverges}[/tex]

if you replace diverges with converges, then it is true however...
 

Related to Determine whether the series converges or diverges.

1. What is the definition of convergence and divergence in a series?

In a series, convergence refers to the property of the terms in the series getting closer and closer to a specific value as more terms are added. Divergence, on the other hand, means that the terms in the series do not approach a specific value but instead increase or decrease without bound.

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

There are several methods for determining convergence or divergence of a series, such as the comparison test, the ratio test, and the root test. These methods involve comparing the given series to known convergent or divergent series or evaluating the limit of the series' terms.

3. What is the importance of determining convergence or divergence in a series?

Determining convergence or divergence in a series is important because it tells us whether the series has a finite sum or not. Convergent series have a finite sum, meaning that the terms in the series add up to a specific value. Divergent series, on the other hand, do not have a finite sum and can either increase or decrease without bound.

4. Can a series converge and diverge at the same time?

No, a series cannot converge and diverge at the same time. A series can only have one of these two properties. If a series is divergent, it means it does not have a finite sum, and therefore, cannot converge. Similarly, if a series is convergent, it cannot diverge.

5. What are some real-life applications of determining convergence or divergence in a series?

Determining convergence or divergence in a series has various real-life applications, such as in finance, engineering, and physics. In finance, it can be used to calculate compound interest or determine the value of investments. In engineering, determining the convergence or divergence of a series can help analyze the stability and behavior of complex systems. In physics, it can be used to model and predict the behavior of natural phenomena.

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