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kd123

  • 2 years ago

Use partial sums to determine if the following series diverge or converge. If the series converges, determine its sum. Σ (from n=0 to infinity) (2/(n^2 + 4n + 3)

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  1. satellite73
    • 2 years ago
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    it definitely converges because the degree of the denominator is 2 and the degree of the numerator is 0 and the difference is greater than 1

  2. satellite73
    • 2 years ago
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    to find the sum, i think the easiest thing to do would be partial fractions do you know how to do that?

  3. kd123
    • 2 years ago
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    yes thank you!

  4. satellite73
    • 2 years ago
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    yw

  5. satellite73
    • 2 years ago
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    once you get the partial fraction, write out the first few terms and you will see that the sum "telescopes" that will not only give you a formula for the partial sums, it will also tell you what the infinite sum is

  6. kd123
    • 2 years ago
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    ok, thanks. Going to try it now.

  7. satellite73
    • 2 years ago
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    good luck!

  8. satellite73
    • 2 years ago
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    write back if you get stuck, i will try it too

  9. kd123
    • 2 years ago
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    For the partial sums I got this

  10. kd123
    • 2 years ago
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    \[\frac{ 1 }{ 2 }+\frac{ 1 }{ 3 }-\frac{ 1 }{n+2 }-\frac{ 1 }{ n+3 }\]

  11. satellite73
    • 2 years ago
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    if you are starting at \(n=0\) isn't the first term 1 ?

  12. kd123
    • 2 years ago
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    Oh yes you are correct. My mistake.

  13. satellite73
    • 2 years ago
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    a small quibble, but it does make a difference i suppose

  14. satellite73
    • 2 years ago
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    i get \(1+\frac{1}{2}\) and everything after that gets killed off

  15. satellite73
    • 2 years ago
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    also i think ( i may be wrong about this) that for the partial sum you get \[\frac{3}{2}-\frac{1}{n+3}\]

  16. kd123
    • 2 years ago
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    yes, I got this as well. Thanks for your help!

  17. satellite73
    • 2 years ago
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    yw

  18. kd123
    • 2 years ago
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    Can you help me again please. How do I go about doing this kind: \[\sum_{n=1}^{\infty}5(\frac{ 2 }{ 3 })^{n-1} \] Use partial sums to determine if that series converges or diverges and if it converges, determine its sum.

  19. satellite73
    • 2 years ago
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    geometric series for this one pull out the 5

  20. kd123
    • 2 years ago
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    ohh

  21. satellite73
    • 2 years ago
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    \[5\sum_{k=0}^{\infty}(\frac{2}{3})^k\] use \[\frac{1}{1-r}\]

  22. satellite73
    • 2 years ago
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    you can almost do it in your head \[1-\frac{2}{3}=\frac{1}{3}\] the reciprocal of \(\frac{1}{3}\) is 3 and you get 15

  23. kd123
    • 2 years ago
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    Thanks you so much.

  24. satellite73
    • 2 years ago
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    yw again partial sums are a pain, but not too bad, it is just \[\frac{1-r^n}{1-r}\] in general

  25. kd123
    • 2 years ago
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    yeah, I'll try to remember that from now on :)

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