Infinite Series: Convergence and Absolute Convergence

In summary, the given series from n = 1 to infinity of An converges to L. Therefore, for the series from n = 1 to infinity of (An)^2, the possible conclusions are that it may diverge, it converges absolutely, it converges to M < L, it converges to M > L, or it converges to M^2 = L. By using the concept of convergent geometric series, we can eliminate options A and D. Additionally, looking up the definition of absolute convergence, we can see that the given series does not converge absolutely, eliminating option B. By playing around with the given series, we can see that it converges to M^2 = L, making option
  • #1
fjotlandj
1
0
Given that the series from n = 1 to infinity of An converges to L, which of the following conclusions is valid for the series from n = 1 to infinity of (An)^2?

A) It may diverge
B) It converges absolutely
C) It converges to M < L
D) It converges to M > L
E) It converges to M^2 = L

My intuition tells me the answer is E). But I am not sure can someone help figure this out please
 
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  • #2
fjotlandj said:
Given that the series from n = 1 to infinity of An converges to L, which of the following conclusions is valid for the series from n = 1 to infinity of (An)^2?

A) It may diverge
B) It converges absolutely
C) It converges to M < L
D) It converges to M > L
E) It converges to M^2 = L

My intuition tells me the answer is E). But I am not sure can someone help figure this out please

Hints: convergent geometric series (this should eliminate some of the possible answers). Then look up what it means to converge absolutely, and play around with

[tex]\sum_{n=1}^\infty \frac{(-1)^{n+1}}{\sqrt{n}}[/tex]
 

FAQ: Infinite Series: Convergence and Absolute Convergence

What is an infinite series?

An infinite series is a sum of an infinite sequence of numbers. It is denoted by the symbol Σ (sigma) and is expressed as a_n = a_1 + a_2 + a_3 + ..., where a_n represents the nth term of the series.

What is the difference between a finite and an infinite series?

A finite series has a finite number of terms, while an infinite series has an infinite number of terms. This means that the sum of a finite series will eventually reach a final value, while the sum of an infinite series may continue on indefinitely.

How can you determine the convergence or divergence of an infinite series?

The convergence or divergence of an infinite series can be determined by applying various convergence tests such as the ratio test, the root test, and the integral test. These tests evaluate the behavior of the terms in the series and determine whether the sum will approach a finite value or diverge to infinity.

Can an infinite series have a finite sum?

Yes, an infinite series can have a finite sum if it converges. This means that the terms in the series become smaller and smaller, approaching zero, and the sum approaches a finite value. However, not all infinite series will have a finite sum, as some may diverge to infinity.

How are infinite series used in real-world applications?

Infinite series are used in various fields of science and engineering, such as physics, economics, and computer science. They are used to model and solve problems involving continuous processes and to approximate complex functions. They are also used in the development of algorithms and in data analysis.

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