- #1
McAfee
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Homework Statement
Show that the Ʃ 1/(2n-1)^3 Converges
The Attempt at a Solution
I tried using the ratio the ratio test but that didn't work
Show that the Ʃ 1/(2n-1)^3 Converges
- Thread starter McAfee
- Start date
In summary, to show that the series Ʃ 1/(2n-1)^3 converges, you can use the comparison test with the zeta(3) series. If needed, you can also use the integral test to establish the convergence of zeta(3). Additionally, the p-series test can also be used to show convergence for p = 3.
Show that the Ʃ 1/(2n-1)^3 Converges
I tried using the ratio the ratio test but that didn't work
- #2
Curious3141
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I assume the summation is over all non-negative n (i.e. 1,2,3..)?
Just compare with zeta(3) (which has all positive terms and includes all the terms in your series), which converges. If you need to establish convergence of the latter, use the integral test, i.e. prove that ∫x^(-n)dx for the bounds [1, infinity) is finite for all n > 1 (here the n is 3).
EDIT: Damn Latex
Just compare with zeta(3) (which has all positive terms and includes all the terms in your series), which converges. If you need to establish convergence of the latter, use the integral test, i.e. prove that ∫x^(-n)dx for the bounds [1, infinity) is finite for all n > 1 (here the n is 3).
EDIT: Damn Latex
Last edited:
- #3
Dick
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Curious3141 said:
Good advice, but it's probably clearer if you say summation 1/n^3 instead of zeta(3). Not everybody knows the Riemann zeta function.
- #4
Curious3141
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Dick said:
Fair enough.
- #5
McAfee
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Dick said:
If I say 1/n^3 could I also use the p-series test.
and yes i meant the summation where n=1 to infinity
- #6
Curious3141
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McAfee said:
If you're allowed to assume the p-series convergence for p = 3, that's essentially the same thing as assuming the convergence for zeta(3). Then you don't even need the integral test. Just compare and you're done.
FAQ: Show that the Ʃ 1/(2n-1)^3 Converges
What does the series Ʃ 1/(2n-1)^3 represent?
The series Ʃ 1/(2n-1)^3 represents an infinite sum of terms where each term is equal to 1 divided by the cube of an odd integer.
What is the mathematical notation for convergence?
The mathematical notation for convergence is lim n→∞ Ʃ a_n = L, which means that the limit of the sum of the terms approaches a finite value L as n approaches infinity.
How do you determine if a series converges?
A series converges if the limit of the sum of its terms approaches a finite value as the number of terms approaches infinity. This can be determined by using convergence tests such as the comparison test, ratio test, or integral test.
What is the convergence test used to show that Ʃ 1/(2n-1)^3 converges?
The convergence test used to show that Ʃ 1/(2n-1)^3 converges is the p-series test, which states that if the series Ʃ a_n converges, then the series Ʃ 1/n^p also converges, where p > 0.
How do you prove that Ʃ 1/(2n-1)^3 converges?
To prove that Ʃ 1/(2n-1)^3 converges, we can use the p-series test with p = 3. This is because the series Ʃ 1/n^3 also converges, and Ʃ 1/(2n-1)^3 is a sub-series of Ʃ 1/n^3 with only the odd terms. Therefore, Ʃ 1/(2n-1)^3 must also converge.