Help with determining if a series is convergent or divergent question

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Homework Help Overview

The discussion revolves around determining the convergence or divergence of a series, with participants exploring the characteristics of the series and the methods for proving convergence.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants express uncertainty about the appropriate method to prove convergence and question whether the series can be classified as geometric. There is discussion about the behavior of the ratio of consecutive terms and its implications for convergence.

Discussion Status

Some participants have offered insights regarding the nature of the series and the limits involved, while others are seeking clarification on the conditions for convergence. Multiple interpretations of the series' behavior are being explored without a clear consensus.

Contextual Notes

Participants are working with a series that is not geometric, as indicated by the non-constant ratio of consecutive terms. There is a focus on the limit of the series as it approaches infinity, with specific limits being discussed but not resolved.

student93
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Homework Statement



Problem is attached in this post.

Homework Equations



Problem is attached in this post.

The Attempt at a Solution



I know how to find the sum of this series, but I don't know what method to use in order to prove that this series converges. Also I don't understand how I'm supposed to tell as to whether or not this is a geometric series.
 

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student93 said:

Homework Statement



Problem is attached in this post.

Homework Equations



Problem is attached in this post.

The Attempt at a Solution



I know how to find the sum of this series, but I don't know what method to use in order to prove that this series converges. Also I don't understand how I'm supposed to tell as to whether or not this is a geometric series.

It's not geometric because the ratio of consecutive terms isn't a constant, but that ratio does approach a limit. Is that enough of a clue?
 
Dick said:
It's not geometric because the ratio of consecutive terms isn't a constant, but that ratio does approach a limit. Is that enough of a clue?

Would the limit have to be 0 for the series to convergent?

Lim (3^n + 4^n)/(7^n) n -> ∞ = 0 For the series to be convergent? (I actually tried this, but can't seem to find a method to solve for such a limit).
 
Last edited:
student93 said:
Would the limit have to be 0 for the series to convergent?

Lim (3^x + 4^x)/(7^x) n -> ∞ = 0 For the series to be convergent? (I actually tried this, but can't seem to find a method to solve for such a limit).

You mean Lim (3^n + 4^n)/(7^n) n -> ∞ = 0. You should be able to. Divide numerator and denominator by 7^n. But that's not what I'm talking about. I'm talking about the ratio ##\frac{a_{n+1}}{a_n}##.
 

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