Determine whether the series converge or diverge

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In summary, for question #1, we can use the nth term test for divergence to determine whether the series converges or diverges. After simplifying the expression, we are left with a limit of 0, which means the series could either converge or diverge. For question #2, we can't simply the expression easily and may need to use another test to determine convergence or divergence.
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taekwondo22
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1) Determine whether the series converges or diverges: summation from n=1 to ∞ of (square root of (n+1) - square root of (n-1)) / n. clearly state which test you are using.

2) Determine whether the series converges or diverges: summation from n=1 to ∞ of (1*3*5*... (2n-1)) / (2*5*8*... (3n-1)). clearly state which test you are using.

For question #1, I tried multiplying the top and bottom by square root of (n+1) + square root of (n-1). On the top, the answer simplifies to 2 and on the bottom it simplifies to n multiplied by (square root of (n+1) + square root of (n-1)). I am thinking to divide the top and bottom by n so the limit as n approaches infinity is equal to 0. But by the nth term test for divergence, if the limit is equal to 0, then the series may converge or diverge. This is where I am stuck and can't think of anything else.

For question #2, I am having trouble simplifying the problem. It can't be just (2n-1) / (3n-1) because that would change the whole series.
 
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In the future, questions like these need to be posted in the Homework Help section of the forum. Anyway, before anyone here can help you, you need to show us what you have tried first.
 
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Mod note: thread moved to homework section
 

FAQ: Determine whether the series converge or diverge

What is the definition of a convergent series?

A convergent series is a series in which the sum of its terms approaches a finite limit as the number of terms increases. In simpler terms, it means that the series eventually settles on a specific value rather than growing infinitely.

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, root test, and integral test. These tests involve comparing the series to other known series or using mathematical techniques to evaluate the series.

What is the significance of determining whether a series converges or diverges?

Determining if a series converges or diverges is essential in mathematics and science as it helps us understand the behavior of infinite sequences. It also allows us to accurately calculate the sum or value of a series, which is crucial in many applications, including physics, engineering, and finance.

Can a series both converge and diverge?

No, a series can only either converge or diverge, not both. A series that converges will eventually reach a finite value, while a series that diverges will either grow infinitely or have no limit.

What happens if the series is neither convergent nor divergent?

If a series is neither convergent nor divergent, it is considered to be oscillating or alternating. This means that the series does not approach a finite value or grow infinitely, but rather fluctuates and does not have a definite behavior. In such cases, further analysis or more advanced mathematical techniques may be needed to determine the behavior of the series.

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