# Determining if a series converges

• Sunwoo Bae
In summary, the conversation discusses using the Limit Comparison Test to determine the convergence of a series with terms ##(\frac 5 4)^n##. The answer sheet claims that the series diverges, which confuses the person asking for help. Through further discussion, it is revealed that the person's mistake was not realizing that the series is actually a divergent geometric series. The Nth Term Test for Divergence is suggested as another method to confirm the divergence of the series.
Sunwoo Bae
Homework Statement
Does the following series converge? Give reasons. (Series shown below)
Relevant Equations
None

The following is my attempt at the solution.
Here, I used limit comparison test to arrive at the answer that the series converges.
However, the answer sheet reads that the series diverges.
I am confused because I cannot figure where my work went wrong…
can anyone tell me how the series diverges, and why my work is incorrect?

Thank you!

The geometric series with terms ##(\frac 5 4)^n## clearly diverges. The formula for a convergent geometric series does not apply. And, in particular, the sum of the series in not ##-5##. That would be too absurd!

PeroK said:
The geometric series with terms ##(\frac 5 4)^n## clearly diverges. The formula for a convergent geometric series does not apply. And, in particular, the sum of the series in not ##-5##. That would be too absurd!
Got it! Thank you!

berkeman
PeroK said:
That would be too absurd!
As opposed to just absurd enough?

Sunwoo Bae said:
Homework Statement:: Does the following series converge? Give reasons. (Series shown below)
Relevant Equations:: None

However, the answer sheet reads that the series diverges.
I am confused because I cannot figure where my work went wrong…
In your work for the Limit Comparison Test, you arrived at a limit of 1. Your mistake was not realizing that ##\sum \frac {5^n}{4^n}## is a divergent series. Since the limit you calculated was positive and finite, the series you were working with had the same behavior as ##\sum \frac {5^n}{4^n}##.

Another test that is sometimes useful is the Nth Term Test for Divergence. Since ##\lim_{n \to \infty} \frac {5^n}{4^n + 3} = \infty## (work not shown), then the series ##\sum_{n \to \infty} \frac {5^n}{4^n + 3}## diverges. Any time you get a limit that isn't 0 or fails to exist in this test, the series diverges.

## 1. What is the definition of a convergent series?

A convergent series is a series in which the sum of its terms approaches a finite limit as the number of terms increases.

## 2. How do you determine if a series converges?

One way to determine if a series converges is by using the limit comparison test, where the limit of the series is compared to the limit of a known convergent or divergent series. Another method is using the ratio test, where the ratio of consecutive terms is compared to a value to determine convergence or divergence.

## 3. What is the difference between absolute and conditional convergence?

Absolute convergence refers to a series where the absolute value of each term converges, while conditional convergence refers to a series where the absolute value of each term may not converge, but the series as a whole still converges.

## 4. Can a series converge if its terms do not approach zero?

Yes, a series can still converge even if its terms do not approach zero. This is known as the alternating series test, where the series alternates between positive and negative terms and the absolute value of the terms decreases as the series progresses.

## 5. What is the significance of the divergence test?

The divergence test is used to determine if a series diverges by checking if the limit of the series is equal to zero. If the limit is not equal to zero, then the series must diverge. However, if the limit is equal to zero, the series may still converge or diverge, and further tests must be used to determine its convergence.

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