Determine whether the series is convergent or divergent

In summary, the conversation is discussing how to determine whether the series ∑ from n=1 to infinity 2 / n(2n+2)^(1/4) is convergent or divergent. The two methods being considered are the P-Series test and the Integral test, and it is suggested to use the comparison test for the former and to integrate for the latter.
  • #1
Hypnos_16
153
1

Homework Statement



I have to find whether the following is Convergent or Divergent

∑ from n = 1 to infinity
2 / n(2n + 2)^(1/4)

Actually it's the fourth root, this is just easier to write.

Homework Equations



According to the front of the sheet it's a quiz on P-Series and Integral Test
I'm leaning more towards Integral Test.

The Attempt at a Solution



Not sure how to go about it, i think I've been looking at them for too long, i can't seem to remember how to do integrals anymore
 
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  • #2
I wouldn't use the integral test.
 
  • #3
Either one will work. The "P-series" test says that a series of the form [itex]\sum n^p[/itex] will converge if p< 1, diverge if [itex]p\ge 1[/itex]. This is not exactly of that form, but you can use the comparison test also. Can you find a p so that [itex]1/(n(2n+2)^{1/4}< n^p[/itex]? The integral test says that a series [itex]\sum a_n[/itex] converges if the integral [itex]\int_1^\infty a(x)dx[/itex] converges where "a(x)" is just [itex]a_n[/itex] with "n" replaced by "x". Can you integrate
[tex]\int_1^\infty \frac{1}{x(x+1)}dx[/tex]?
 

What is the purpose of determining whether a series is convergent or divergent?

The purpose of determining whether a series is convergent or divergent is to understand the behavior of the series as the number of terms increases. It helps to determine if the sum of the series approaches a finite value (convergent) or if it diverges to infinity.

What is the difference between a convergent and a divergent series?

A convergent series is one where the sum of the terms approaches a finite value as the number of terms increases. In a divergent series, the sum of the terms does not approach a finite value and may either increase or decrease without bound.

What are some common tests used to determine the convergence or divergence of a series?

Some common tests used to determine the convergence or divergence of a series include the comparison test, ratio test, and integral test. These tests compare the given series to a known convergent or divergent series to determine the behavior of the given series.

What is the importance of understanding the convergence or divergence of a series?

Understanding the convergence or divergence of a series is important in many areas of mathematics and science. It allows us to make predictions and draw conclusions about the behavior of a system or process. It also helps us to determine the accuracy of our calculations and to know when a series may be used to approximate a value.

Can a series be both convergent and divergent?

No, a series can only be either convergent or divergent, not both. If the sum of a series approaches a finite value, it is considered to be convergent. If the sum of the series does not approach a finite value, it is considered to be divergent.

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