Real Analysis Question regarding Series

In summary, the radius of convergence for the power series \suma_nx^n is at least 1, given that the sequence {a_n} is monotonically decreasing and has a limit of 0. The proof involves using the definition of the radius of convergence and showing that the limit of the ratio of consecutive terms in the sequence is at least 1. The fact that a_n approaches 0 as n approaches infinity is not necessary for this proof.
  • #1
christianrhiley
4
0

Homework Statement



Let {a_n} be a monotonically decreasing sequence of positive real numbers with lim a_n = 0. Show the radius of convergence of [tex]\sum[/tex]a_nx[tex]^{}n[/tex] is at least 1.


The Attempt at a Solution


I have no real attempt at a solution since I'm unsure how to proceed. I've tried using the power series definition and using a fixed x_0, but I get nowhere using this method. Can someone outline how this proof might look? Thanks.
 
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  • #2
The radius of convergence is given by
[tex]\lim_{n \to \infty} \left| \frac{a_n}{a_{n+1}} \right|.[/tex]

If {a_n} is decreasing and lim a_n = 0, then what can we say about this limit?
 
  • #3
Is the fact that [itex]a_n = 0[/itex] as n approaches infinity needed? I find no reason for it.
 
  • #4
Solved It!

Solved It!
 

Related to Real Analysis Question regarding Series

What is a series in real analysis?

A series is a sequence of numbers that are added together. In real analysis, a series is typically an infinite sum of terms, and the goal is to determine whether the series converges or diverges.

What is the difference between a convergent and divergent series?

A convergent series is one in which the sum of the terms approaches a finite limit as the number of terms increases. In contrast, a divergent series is one in which the sum of the terms does not approach a finite limit and instead continues to increase or decrease without bound.

How do you determine the convergence or divergence of a series?

In real analysis, there are several tests that can be used to determine the convergence or divergence of a series. These include the comparison test, ratio test, root test, and integral test. These tests involve comparing the given series to a known series or using mathematical operations to analyze the behavior of the terms in the series.

What is the limit comparison test and when is it used?

The limit comparison test is a method for determining the convergence or divergence of a series by comparing it to a known series. It is used when the terms in the given series are difficult to analyze or do not fit into any of the other commonly used tests.

Can a series converge conditionally but not absolutely?

Yes, it is possible for a series to converge conditionally but not absolutely. This means that the series converges, but if the terms of the series are rearranged, the sum of the series will change. In contrast, a series that converges absolutely will have the same sum regardless of the order in which the terms are added.

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