How do you use the integral test to determine if #Sigma n/(n^4+1)# from #[1,oo)# is convergent or divergent?

Answer 1
The series #sum_(n=N)^ooa_n# can be tested via the integral test if #a_n=f(x)# is:
We see that #f(x)=x/(x^4+1)# fits both of these on #x in [1,oo)#. The integral test then states that if the integral
#int_N^oof(x)dx#
is finite, or if the interval converges to any value, then the series #sum_(n=N)^ooa_n# converges as well.

If the integral diverges, so does the series.

So, we want to test the convergence of the integral:

#int_1^oox/(x^4+1)dx=1/2int_1^oo(2x)/((x^2)^2+1)dx#
Let #u=x^2#:
#color(white)(int_1^oox/(x^4+1)dx)=1/2int_1^oo1/(u^2+1)du#
#color(white)(int_1^oox/(x^4+1)dx)=1/2arctan(u)|_1^oo#
Note that #lim_(urarroo)arctan(u)=pi/2# and #arctan(1)=pi/4#:
#color(white)(int_1^oox/(x^4+1)dx)=1/2(pi/2-pi/4)=pi/8#

Since this integral converges, so does the given series via the integral test.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To use the integral test to determine the convergence or divergence of the series (\sum_{n=1}^{\infty} \frac{n}{n^4 + 1}), you need to follow these steps:

  1. Evaluate the corresponding improper integral: [ \int_{1}^{\infty} \frac{x}{x^4 + 1} , dx ]

  2. Determine if the integral converges or diverges.

  3. If the integral converges, then the series (\sum_{n=1}^{\infty} \frac{n}{n^4 + 1}) converges. If the integral diverges, then the series also diverges.

So, to determine convergence or divergence, evaluate the integral (\int_{1}^{\infty} \frac{x}{x^4 + 1} , dx) and ascertain its convergence or divergence.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7