## 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

|dw:1437979024641:dw|

2. ganeshie8

$\large \sum\limits_{n=1}^{\infty}~\dfrac{7}{n^2+25}$ like that ?

3. andu1854

yes

4. ganeshie8

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

so evaluate the integral and see if it converges (finite)

6. Astrophysics

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

Hint: $\int\limits \frac{ 1 }{ x^2+a^2 } dx = \frac{ 1 }{ 2 } \tan^{-1}\left( \frac{ x }{ a } \right)+C$

8. andu1854

Ok so I understand that this will converge: so would it be 1/2 tan ^-1 (1/5) then?

9. anonymous

yes

10. andu1854

ok then would I multiply this by 7 because 7 is the constant?

11. Zale101

Correct.

12. andu1854

ok I did this and got .69 ( rounded to nearest 100th)... so what would I do next...

13. Astrophysics

No, your result is wrong, it's an improper integral..

14. Astrophysics

$\int\limits_{0}^{\infty} \frac{ 7 }{ x^2+25 } dx$

15. andu1854

Ok its positive, it is continuous and it is decreasing as we move towards infinity... so:|dw:1437984135224:dw|

16. andu1854

|dw:1437984288640:dw|

17. andu1854

because the series starts at n =1, I felt a should be = 1 and b = inifinity