Understanding the Ratio Test for Series and Its Applications

In summary, the conversation discusses the use of the ratio test for a series with a given initial term and recursive formula. The approach of using the ratio test is explored, but a simpler solution is suggested. The limit is then evaluated and it is determined that the series converges.
  • #1
18
8
Homework Statement
Use the ratio test for the series ##\sum_{n=1}^{\infty} b_{n}## where ##b_{1} = 5## and ##b_{n}= \frac {(-1)^{n}nb_{n-1}}{3n+1}## for ##n \geq 2##
Relevant Equations
ratio test: ##L = \lim_{n \to \infty} \lvert \frac {a_{n+1}}{a_{n}} \rvert## will converge if ##L < 1## and will diverge if ##L > 1##
So I am having some difficulty expressing this series explicitly. I just tried finding some terms

##b_{0} = 5##

I am assuming I am allowed to use that for ##b_{1}## for the series, even if the series begins at ##n=1##? With that assumption, I have

##b_{1} = -\frac {5}{4}##
##b_{2} = - \frac{5}{14}##
##b_{3} = \frac {3}{28}##
## b_{4} = \frac {3}{91}##

I am not seeing a pattern. I also alternatively tried using the ratio test not in explicit terms

##\lim_{n \to \infty} \lvert \frac {(-1)^{n+1}(n+1)b_{n}}{3(n+1)+1} \cdot \frac {3n+1}{(-1)^{n}nb_{n-1}} \rvert## and after substituting I ended up with ##\lvert \frac{b_{n}}{b_{n-1}} \rvert## which to me is inconclusive.
 
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  • #2
woopydalan said:
Homework Statement:: Use the ratio test for the series ##\sum_{n=1}^{\infty} b_{n}## where ##b_{1} = 5## and ##b_{n}= \frac {(-1)^{n}nb_{n-1}}{3n+1}## for ##n \geq 2##
Relevant Equations:: ratio test: ##L = \lim_{n \to \infty} \lvert \frac {a_{n+1}}{a_{n}} \rvert## will converge if ##L < 1## and will diverge if ##L > 1##

So I am having some difficulty expressing this series explicitly. I just tried finding some terms
You're making this way too complicated.
$$b_{n}= \frac {(-1)^{n}nb_{n-1}}{3n+1} \quad \Rightarrow \quad \frac{b_{n}}{b_{n-1}}= \frac {(-1)^{n}n}{3n+1}$$
 
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Likes woopydalan
  • #3
Oh wow...let me try that
 
  • #4
Ok, so now I got ##\lim_{n \to \infty} \frac {n}{3n+1}## which would be indeterminate??
 
  • #5
Do you mean indeterminate in the sense that the limit is ##\infty/\infty## or do you mean the convergence of the series can't be determined?
 
  • #6
I think in both senses
 
  • #7
Hint:
[tex]\lim_{n \to \infty} \frac{an+b}{cn+d} = \lim_{n \to \infty} \frac{a + \frac{b}{n}}{c + \frac{d}{n}} = \frac{a}{c}[/tex] for [itex]c \neq 0[/itex].
 
  • #8
Alright, so 1/3, so L < 1, so converges
 

What is the ratio test for series?

The ratio test for series is a mathematical test used to determine the convergence or divergence of a series. It involves taking the limit of the ratio of consecutive terms in the series and comparing it to a critical value.

How is the ratio test applied?

The ratio test is applied by taking the limit of the ratio of consecutive terms in a series. If the limit is less than 1, the series is convergent. If the limit is greater than 1, the series is divergent. If the limit is equal to 1, the test is inconclusive and another test must be used.

What are the applications of the ratio test?

The ratio test has many applications in mathematics, including determining the convergence of infinite series, evaluating limits of functions, and proving the convergence of improper integrals.

When should the ratio test be used?

The ratio test should be used when the series being tested has terms that involve powers or factorials, or when the comparison test is inconclusive. It is also useful for determining the radius of convergence for power series.

What are the limitations of the ratio test?

The ratio test can only be used for series with positive terms. It also cannot determine the convergence or divergence of alternating series. In some cases, the limit of the ratio may be difficult to compute, making the test impractical to use.

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