Finding the Limit of a Series: Exploring Different Patterns and Approaches

In summary, the conversation revolves around finding the limit of a given series, ##S_n##=##\dfrac {3}{8}##⋅##\dfrac {4^n}{3^n}##, and whether it is correct to state ##∞## as the limit. The speaker also mentions trying to use L'Hopital's rule and getting a result of ##∞##, implying divergence. However, another speaker points out that the formula is wrong and may only fit the first two values. There is also a discussion about another series, ##S_n##=##\dfrac {n}{1+n}##, and its pattern.
  • #1
chwala
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Homework Statement
See attached.
Relevant Equations
Limits
Consider the series below;

1644709958718.png


From my own calculations, i noted that this series can also be written as ##S_n##=##\dfrac {3}{8}##⋅##\dfrac {4^n}{3^n}##. If indeed that is the case then how do we find the limit of my series to realize the required solution of ##1## as indicated on the textbook? I tried taking limits...L Hopital's rule... and still got ##∞## implying divergence. Are we required to solely maintain the series in the pattern indicated on the text and not any other way?

In other words, if one was given a specific question to find the limit of my series ##S_n##=##\dfrac {3}{8}##⋅##\dfrac {4^n}{3^n}##in an exam, then would it be correct to state ##∞##?
 
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  • #2
Your formula is wrong. You probably fit it to the first two values, and didn't check if it matches anything else.
 
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Likes chwala
  • #3
Office_Shredder said:
Your formula is wrong. You probably fit it to the first two values, and didn't check if it matches anything else.
True it is wrong...i ought to have checked if it applies to the sequence ##S_3##...but we could also have
...our series ##\dfrac {1}{2}##, ##\dfrac {2}{3}##, ##\dfrac {3}{4}##,##\dfrac {4}{5},##... as
##S_n##=##\dfrac {n}{1+n}##
Cheers :cool:
 
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1. What is the purpose of finding the limit of a series?

The limit of a series is used to determine the behavior of the series as the number of terms approaches infinity. It helps to determine if the series converges or diverges.

2. How do you find the limit of a series?

To find the limit of a series, you can use various techniques such as the ratio test, comparison test, or the root test. These tests help to determine the convergence or divergence of the series.

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

A convergent series has a finite limit, meaning that the sum of the terms in the series approaches a specific value as the number of terms increases. On the other hand, a divergent series does not have a finite limit and the sum of the terms either increases or decreases without approaching a specific value.

4. Can a series have more than one limit?

No, a series can only have one limit. If a series has more than one limit, it is considered divergent.

5. Why is finding the limit of a series important in mathematics?

Finding the limit of a series is important in mathematics because it helps to determine the convergence or divergence of a series. This information is crucial in many mathematical applications, such as in calculus, where series are used to represent functions and make approximations.

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