`1/2 + 1/5 + 1/10+1/17+1/26+...` Confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series.

Expert Answers
gsarora17 eNotes educator| Certified Educator


The series can be written as,


So based on the above pattern we can write the series as,


The integral test is applicable if f is positive, continuous and decreasing function on the infinite interval `[k,oo)` where `k>=1` and `a_n=f(x)` . Then the series converges or diverges if and only if the improper integral `int_k^oof(x)dx` converges or diverges.

For the given series `a_n=1/(n^2+1)`

Consider `f(x)=1/(x^2+1)`


Graph of the function is attached. From the graph we can see that the function is positive, continuous and decreasing on the interval `[1,oo)`

Since the function satisfies the conditions for the integral test, we can apply integral test.

Now let's determine whether the corresponding improper integral `int_1^oo1/(x^2+1)dx` converges or diverges.


Use the common integral: `int1/(x^2+1)dx=arctan(x)`






Since the integral `int_1^oo1/(x^2+1)dx` converges, we conclude from the integral test that the series also converges.

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