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Study23

  • 2 years ago

Okay. I'm having trouble with series. Here's the problem (I'm looking to see whether the series is convergent or divergent, if convergent I have to find the sum it converges to): summation from n=1 to infinity of \(\ \frac{n}{n+5} \).

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  1. Study23
    • 2 years ago
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    I'm thinking partial fractions or factoring n out but I'm not sure...

  2. macknojia
    • 2 years ago
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    can you type this out i can't read it. Sorry :(

  3. Study23
    • 2 years ago
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    ?

  4. macknojia
    • 2 years ago
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    nvm, i got it. You can view this as a geometric series.\[n/(n-(-5))\]

  5. macknojia
    • 2 years ago
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    so your first term is n and your ratio is -5. now using the formula you can find the sum it converges to.

  6. Study23
    • 2 years ago
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    So you factor the n out?

  7. Study23
    • 2 years ago
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    ???

  8. macknojia
    • 2 years ago
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    not really. are you familiar with the formula to find the sum of a convergent series ?

  9. Study23
    • 2 years ago
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    I meant to find what a and r is before I find the sum.

  10. Study23
    • 2 years ago
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    Because r is less than 1 (-5) does that mean this is divergent?

  11. tkhunny
    • 2 years ago
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    What is the limit of the TERMS as n increases without bound? If it's not zero, it doesn't converge.

  12. Study23
    • 2 years ago
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    But r has to be -1<r<1 right?

  13. Study23
    • 2 years ago
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    r is -5 so...

  14. tkhunny
    • 2 years ago
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    Yes, that is good for 'r', but you don' even need that. Just look at the limit of the terms. Test #1, in ALL cases.

  15. tkhunny
    • 2 years ago
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    BTW, r is NOT 5. It's 1.

  16. Study23
    • 2 years ago
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    . ? And wouldn't the limit as n approaches infinity be 0 because denominator is larger?

  17. tkhunny
    • 2 years ago
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    No. The limit of the terms is 1. As n increases, the 5 becomes insignificant.

  18. Study23
    • 2 years ago
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    Okay. And how do I find r to be one then?

  19. macknojia
    • 2 years ago
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    the coefficient of numerator and the denominator is 1 therefore, the limit is one.

  20. macknojia
    • 2 years ago
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    try graphing it.

  21. tkhunny
    • 2 years ago
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    First, I don't actually care. Since the limit of the terms is 1, it cannot converge. There is no reason to do the ration test. Second, just do it. \(\dfrac{\dfrac{n+1}{n+6}}{\dfrac{n}{n+5}} = \dfrac{n^{2}+6n+5}{n^{2}+6n}\). The limit of this, as n increases without bound, is 1. As n increases, 5 becomes insignificant and it is clearly 1.

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