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bahrom7893

  • 2 years ago

For what values of r>0 does the series converge?

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  1. bahrom7893
    • 2 years ago
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    I'm wary of using the integral test before actually getting some advices: |dw:1330831400158:dw|

  2. .Sam.
    • 2 years ago
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    yeah

  3. bahrom7893
    • 2 years ago
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    .Sam. yeah what? Sorry to sound rude, but what's the point of disappointing me? I got so happy when I saw that notification pop up.

  4. bahrom7893
    • 2 years ago
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    @Mertsj can u take a look?

  5. .Sam.
    • 2 years ago
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    uh I got it from a calculator it says that its converge but I don't know how it solves, lol

  6. bahrom7893
    • 2 years ago
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    @malevolence19 can u help?

  7. .Sam.
    • 2 years ago
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    sry for the disappointment :D

  8. bahrom7893
    • 2 years ago
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    tnx for tryin sam

  9. malevolence19
    • 2 years ago
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    Okay, I think I can help you.

  10. bahrom7893
    • 2 years ago
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    awesome

  11. malevolence19
    • 2 years ago
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    For convergence of a series using the ratio test we must have that: \[\lim_{n \rightarrow \infty}a_n \rightarrow 0\] And: \[\lim_{n \rightarrow \infty} \left| \frac{a_{n+1}}{a_n}\right|<1\] So: \[\lim_{n \rightarrow \infty} \left| \frac{r^{\ln(n+1)}}{r^{\ln(n)}}\right|=\lim_{n \rightarrow \infty} \left| r^{\ln(n+1)-\ln(n)}\right|=\lim_{n \rightarrow \infty} \left| r^{\ln \left( \frac{n+1}{n}\right)}\right| \rightarrow 1\] Means that the ratio test is inconclusive. We need to find another approach (I should have realized this wouldn't work, its a rational function :/)

  12. bahrom7893
    • 2 years ago
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    @JamesJ help?!

  13. JamesJ
    • 2 years ago
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    mal19 almost had it. If |r| < 1, then the limit of the ratio a_{n+1}/a_n converges to zero.

  14. bahrom7893
    • 2 years ago
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    oh so that's it?

  15. bahrom7893
    • 2 years ago
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    It's that simple? dang it and i'm sitting here with improper integrals!

  16. JamesJ
    • 2 years ago
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    *correction: If |r| < 1, then the limit of the ratio a_{n+1}/a_n converges to a number less than 1. And then by that ratio test, the sum converges.

  17. bahrom7893
    • 2 years ago
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    U guys rule!

  18. JamesJ
    • 2 years ago
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    namely, it converges to |r| itself.

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