`sum_(n=1)^oo n/(n^4+1)` 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.

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The integral test is applicable if f is positive, continuous and decreasing function on infinite interval `[k,oo)` where `k>=1` and `a_n=f(x)` . Then the series `sum_(n=1)^ooa_n`  converges or diverges if and only if the improper integral `int_1^oof(x)dx` converges or diverges.

For the given series `a_n=n/(n^4+1)`

Consider `f(x)=x/(x^4+1)`

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

We can determine whether function is decreasing, also ,by finding the derivative f'(x) such that `f'(x)<0` for `x>=1` .

We can apply integral test , since the function satisfies the conditions for the integral test.

Now let's determine whether the corresponding improper integral `int_1^oox/(x^4+1)dx` converges or diverges.


Let's first evaluate the indefinite integral `intx/(x^4+1)dx`

Apply integral substitution:`u=x^2`



Take the constant out and use common integral:`int1/(x^2+1)dx=arctan(x)+C`


Substitute back `u=x^2`








Now `lim_(b->oo)(b^2)=oo`

`=1/2(pi/2)-pi/8` [by applying the common limit:`lim_(u->oo)arctan(u)=pi/2` ]



Since the integral `int_1^oox/(x^4+1)dx` converges, we conclude from the integral test that the series converges.

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