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Pologirl19

Consider the infinite geometric series a. Write the first four terms of the series. b. Does the series diverge or converge? c. If the series has a sum, find the sum.

  • one year ago
  • one year ago

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  1. Pologirl19
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    \[\sum_{x=1}^{\infty}-4\frac{ 1 }{ 3 }^{x-1}\]

    • one year ago
  2. Pologirl19
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    Help!!

    • one year ago
  3. tkhunny
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    It's a geometric series with r < 1. It converges and the sum is well known. \(-4 (1 + 1/3 + 1/9 + 1/27 + ...) = \dfrac{-4}{1-(1/3)}\) Try not to struggle over ones like this.

    • one year ago
  4. Pologirl19
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    wait were do you get 1/9 and 1/27. Because I got b. it converges and c. it is -6. But I have to show my work. @tkhunny

    • one year ago
  5. Pologirl19
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    Because I I know the formula. it is \[S=\frac{ a1 }{ 1-r }\]

    • one year ago
  6. tkhunny
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    \(\sum\limits_{x=1}^{\infty} -4\left(\dfrac{1}{3}\right)^{x-1} = -4\sum\limits_{x=1}^{\infty} \left(\dfrac{1}{3}\right)^{x-1} = -4\left(\left(\dfrac{1}{3}\right)^{1-1} + \left(\dfrac{1}{3}\right)^{2-1} + \left(\dfrac{1}{3}\right)^{3-1} + ...\right)\) Are you yet seeing it?

    • one year ago
  7. Pologirl19
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    Ah thank you so much I was really confused. except for I don't get a.

    • one year ago
  8. tkhunny
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    (1/3)^(1-1) = (1/3)^0 = 1 -4 * 1 = -4 = a1

    • one year ago
  9. Pologirl19
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    Thats a.?

    • one year ago
  10. tkhunny
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    a1 is the first term in the series. -4 in this case.

    • one year ago
  11. Pologirl19
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    Wait still a little confused thats only one though

    • one year ago
  12. tkhunny
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    a is the first term. -4 Use x = 1 in the summation formula. r is the common ratio. 1/3 Just read it from the summation formula. \(\dfrac{-4}{1-1/3} = \dfrac{-4}{2/3}\) You're not going to make me do ALL the work, are you?

    • one year ago
  13. Pologirl19
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    No sorry. So it would be -4, 1/3, 1 and infinity. Right

    • one year ago
  14. tkhunny
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    What would be -4? What would be 1/3? What would be 1? What would be "infinity"? Its an infinite series. Its first term is -4 Its common ration is 1/3 That is ALL the required information to find the sum.

    • one year ago
  15. Pologirl19
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    Well it says write the first four terms of the series

    • one year ago
  16. tkhunny
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    Then do that. Substitute x = 1, 2, 3, 4. All should be negative. All should be finite.

    • one year ago
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