# How do you use the integral test to determine whether the following series converge of diverge #sum n/((n^2+1)^2)# from n=1 to infinity? Thanks for the help !!! I have no idea on how to do these questions?

The integral test just says, basically:

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

The series converges.

Let

In order to use the intergral test,

Let's study the sign of

We want to use the integral test for n=1 to infinity. Since

To find the integral of

The series converges if

The series converges.

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

To use the integral test to determine the convergence or divergence of the series ∑ (n/((n^2+1)^2)) from n=1 to infinity:

- Calculate the integral of the function f(x) = x/((x^2+1)^2) from 1 to infinity.
- If the integral converges, then the series also converges. If the integral diverges, then the series also diverges.

Integral calculation:

∫(x/((x^2+1)^2)) dx from 1 to ∞

To evaluate this integral, you can use trigonometric substitution. Let x = tan(u), then dx = sec^2(u) du.

The integral becomes:

∫(tan(u) / (tan^2(u) + 1)^2) * sec^2(u) du from 0 to π/2

After simplifying and integrating, you should obtain a convergent result.

Therefore, since the integral converges, by the integral test, the series ∑ (n/((n^2+1)^2)) from n=1 to infinity also converges.

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

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.

- How do you use the ratio test to test the convergence of the series #∑ 11^n/((n+1)(7^(2n+1)))# from n=1 to infinity?
- How do you use the Comparison Test to see if #1/(4n^2-1)# converges, n is going to infinity?
- What is the difference between an infinite sequence and an infinite series?
- Using the integral test, how do you show whether #sum 1 / [sqrt(n) * (sqrt(n) + 1)]# diverges or converges from n=1 to infinity?
- How do you find the nth partial sum, determine whether the series converges and find the sum when it exists given #1/(1*3)+1/(3*5)+1/(5*7)+...+1/((2n-1)(2n+1))+...#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7