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andu1854

  • one year ago

Use the Integral test to determine if the series shown below converges or diverges. Be sure to check that conditions of the integral test are satisfied. Infinity Sigma n =1 for 7/(n^2 +25) I understand that this does converge, but I need help trying to calculate where this converges to

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  1. andu1854
    • one year ago
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    |dw:1437979024641:dw|

  2. ganeshie8
    • one year ago
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    \[\large \sum\limits_{n=1}^{\infty}~\dfrac{7}{n^2+25}\] like that ?

  3. andu1854
    • one year ago
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    yes

  4. ganeshie8
    • one year ago
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    Integral test tells us this : If \(\int\limits_1^{\infty} \dfrac{7}{x^2+25}\,dx\) converges, then the given series also converges

  5. ganeshie8
    • one year ago
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    so evaluate the integral and see if it converges (finite)

  6. Astrophysics
    • one year ago
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    You also have to check the rules right, \[f(x) = \frac{ 7 }{ x^2+25 }\] is continuous on [1, infinity) then f(x) is positive. You also need to check if f is decreasing on [1, infinity) you can do this by taking the derivative, then if all those agree, you must evaluate the integral.

  7. Astrophysics
    • one year ago
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    Hint: \[\int\limits \frac{ 1 }{ x^2+a^2 } dx = \frac{ 1 }{ 2 } \tan^{-1}\left( \frac{ x }{ a } \right)+C\]

  8. andu1854
    • one year ago
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    Ok so I understand that this will converge: so would it be 1/2 tan ^-1 (1/5) then?

  9. anonymous
    • one year ago
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    yes

  10. andu1854
    • one year ago
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    ok then would I multiply this by 7 because 7 is the constant?

  11. Zale101
    • one year ago
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    Correct.

  12. andu1854
    • one year ago
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    ok I did this and got .69 ( rounded to nearest 100th)... so what would I do next...

  13. Astrophysics
    • one year ago
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    No, your result is wrong, it's an improper integral..

  14. Astrophysics
    • one year ago
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    \[\int\limits_{0}^{\infty} \frac{ 7 }{ x^2+25 } dx \]

  15. andu1854
    • one year ago
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    Ok its positive, it is continuous and it is decreasing as we move towards infinity... so:|dw:1437984135224:dw|

  16. andu1854
    • one year ago
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    |dw:1437984288640:dw|

  17. andu1854
    • one year ago
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    because the series starts at n =1, I felt a should be = 1 and b = inifinity

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