Help with determining if a series is convergent or divergent question

In summary, the conversation discusses a problem attached in the post and the attempt to find the sum of the series and prove its convergence. It is stated that the series is not geometric, but the ratio of consecutive terms approaches a limit. The question arises whether the limit has to be 0 for the series to be convergent, and the suggestion is made to divide the numerator and denominator by 7^n to find the ratio. However, the main focus is on the ratio ##\frac{a_{n+1}}{a_n}## and its role in determining convergence.
  • #1
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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  • #2
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?
 
  • #3
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:
  • #4
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}##.
 

1. What is the definition of a convergent series?

A convergent series is a sequence of numbers where the terms approach a finite limit as the number of terms increases. In other words, the sum of all the terms in a convergent series will eventually reach a finite value.

2. How do you test for convergence of a series?

There are several methods for testing the convergence of a series, such as the comparison test, the ratio test, and the integral test. These methods involve comparing the given series to a known convergent or divergent series or using calculus to evaluate the limit of the series.

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

A convergent series has a sum that approaches a finite value, while a divergent series has a sum that either approaches infinity or does not approach any value at all. In other words, a convergent series has a finite sum, while a divergent series has an infinite sum.

4. Can a series be both convergent and divergent?

No, a series can only be either convergent or divergent, but not both. A series cannot have a finite and an infinite sum at the same time.

5. How does the behavior of the terms in a series affect its convergence or divergence?

The behavior of the terms in a series, such as their growth rate or whether they alternate in sign, can determine whether the series is convergent or divergent. For example, a series with terms that decrease in value will likely be convergent, while a series with terms that increase in value will likely be divergent.

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