Absolute & conditional convergence

• magnifik
In summary, the conversation involves determining whether two given series converge absolutely, conditionally, or not at all. The first series involves a comparison test and the second series is absolutely convergent. The confusion between n's and x's in the conversation is noted and addressed.

Homework Statement

Determine whether the series converges absolutely, conditionally, or not at all.

a) $$\Sigma$$ (-1)nn4/(x3 + 1)

b) $$\Sigma$$ sin(x)/x2

The Attempt at a Solution

a) positive series is n4/n3+1 .. do i do comparison test ??

b) |sin(x)|/x2
compare it with 1/x2 which converges.. so it's absolutely convergent??

You are kind of freely mixing n's and x's here. Are they supposed to be the same? If so, for the first one ask whether the nth term goes to zero. For the second one, yes, it's absolutely convergent.

woops, mixing up the n's & x's was a careless mistake

for a) the positive series diverges because n^4/n^3 + 1 goes to infinity, but I'm not sure if the original series converges or diverges

magnifik said:
for a) the positive series diverges because n^4/n^3 + 1 goes to infinity, but I'm not sure if the original series converges or diverges

A series whose terms don't go to zero diverges no matter what the signs on the terms. Look at the definition of convergence in terms of partial sums.

1. What is the difference between absolute and conditional convergence?

Absolute convergence refers to a series or sequence where the sum of the absolute values of its terms is finite. Conditional convergence, on the other hand, refers to a series or sequence where the sum of its terms is finite, but the sum of the absolute values of its terms is infinite.

2. How do you determine if a series or sequence is absolutely convergent?

A series or sequence is absolutely convergent if the limit of the absolute value of its terms approaches zero as the number of terms approaches infinity. This can be determined using the ratio test, comparison test, or integral test.

3. Can a conditionally convergent series or sequence be rearranged to have a different sum?

Yes, a conditionally convergent series or sequence can be rearranged to have a different sum. This is known as the Riemann rearrangement theorem.

4. What are some real-world applications of absolute and conditional convergence?

Absolute and conditional convergence are important concepts in mathematics, particularly in the study of series and sequences. They also have applications in engineering, physics, and economics, such as in the analysis of alternating current circuits, Fourier series, and Taylor series.

5. How do you use the alternating series test to determine convergence?

The alternating series test can be used to determine the convergence of an alternating series, where the terms alternate in sign. The test states that if the absolute value of the terms decreases and approaches zero as the number of terms approaches infinity, then the series is convergent. However, this test does not determine absolute convergence.