Convergence problem (nth-term test)

In summary, to show that the sum of (n/(3n+1))n from n=1 to ∞ converges, the book uses a comparison test to (1/3)n, but an n-th term test somewhere may result in a mistake. The function an = (n/(3n+1))n can be rewritten as an = e(-1/3), which would imply divergence. However, the series actually converges because if an does not converge to 0, the series diverges. This means that the series may or may not converge if an converges to a real number other than 0. The mistake in the conversation lies in using l'Hopital's rule, as the top function,
  • #1
tolove
164
1
Show that the sum of (n/(3n+1))n from n=1 to ∞ converges.

The book solves this with a comparison test to (1/3)n, but I'm making a mistake with an n-th term test somewhere.

an = (n/(3n+1))n
Take ln of both sides, then use n = 1/(1/n) to setup for l'Hopital's rule.
ln an = ln(n/(3n+1)) / (1/n)
l'Hop
ln an = -n/(3n+1)
l'Hop
ln an = -1/3
raise both sides
an = e(-1/3)

Which would mean divergence, right? Since the nth term does not equal 0?

But this problem converges
 
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  • #2
tolove said:
Show that the sum of (n/(3n+1))n from n=1 to ∞ converges.

The book solves this with a comparison test to (1/3)n, but I'm making a mistake with an n-th term test somewhere.

an = (n/(3n+1))n
Take ln of both sides, then use n = 1/(1/n) to setup for l'Hopital's rule.
ln an = ln(n/(3n+1)) / (1/n)
l'Hop
But that is neither an ##\frac\infty \infty## or ##\frac 0 0## form.
 
  • #3
LCKurtz said:
But that is neither an ##\frac\infty \infty## or ##\frac 0 0## form.

Ohh, ok

1) If [itex]\sum[/itex] an converges, then an → 0

2) [itex]\sum[/itex] an may or may not converge if an → c, c being any real number.

3) If an as n → ∞ fails to exist, then [itex]\sum[/itex] an diverges.

e: thank you very much, this was driving me nuts
 
  • #4
tolove said:
Ohh, ok

1) If [itex]\sum[/itex] an converges, then an → 0

2) [itex]\sum[/itex] an may or may not converge if an → c, c being any real number.

If ##a_n## does anything but converge to 0, the series diverges. If it does converge to zero, the series may or may not converge. That's all there is to it.
 
  • #5
LCKurtz said:
If ##a_n## does anything but converge to 0, the series diverges. If it does converge to zero, the series may or may not converge. That's all there is to it.

Double thanks, my mistake is with l'Hopital's rule.

f(a) = g(a) = 0 as the limit of n → a must be true.
The top function:
ln(n/(3n+1))
does not go to 0 as n → infinity, so l'Hopital's rule cannot be applied.
 

FAQ: Convergence problem (nth-term test)

1. What is the convergence problem (nth-term test)?

The convergence problem, also known as the nth-term test, is a mathematical test used to determine whether an infinite series converges or diverges. It involves evaluating the limit of the nth term of the series as n approaches infinity. If the limit is equal to zero, the series converges. If the limit is non-zero or does not exist, the series diverges.

2. How is the convergence problem (nth-term test) used in real-world applications?

The convergence problem is used in various fields such as physics, engineering, and economics to determine the convergence or divergence of series that represent real-world phenomena. For example, it can be used to analyze the stability of a system or to predict the future behavior of a variable based on its past values.

3. What are the limitations of the convergence problem (nth-term test)?

The convergence problem is only applicable to infinite series, and it can only determine the convergence or divergence of a series, not its actual sum. It also cannot be used for alternating series or series with complex terms.

4. What is the relationship between the convergence problem (nth-term test) and other convergence tests?

The convergence problem is one of the most commonly used convergence tests, but it is not always the most efficient or conclusive. Other tests, such as the ratio test and the integral test, may provide more information about the behavior of a series. The convergence problem is often used in conjunction with other tests to confirm the convergence or divergence of a series.

5. How does the convergence problem (nth-term test) relate to the concept of convergence in calculus?

In calculus, convergence refers to the behavior of a sequence or series as its terms get closer and closer to a certain value. The convergence problem uses this concept to determine whether a series converges or diverges. If the terms of a series approach a certain value, the series is said to converge. If the terms do not approach a specific value, the series diverges.

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