Using the integral test, how do you show whether #n/(n^2+1)# diverges or converges?

Answer 1
#int_1^oo x/(x^2+1) dx#
Let #u=x^2+1#, so that #du= 2x dx#
#int x/(x^2+1) dx = 1/2 int (2x)/(x^2+1) dx = 1/2 1/u du = 1/2 ln abs(u) +C#
#int_1^oo x/(x^2+1) dx =lim_(brarr oo) 1/2ln(x^2+1)|_1^b # which diverges,

So the series diverges.

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Answer 2

To determine whether the series ( \sum_{n=1}^{\infty} \frac{n}{n^2 + 1} ) converges or diverges using the integral test, we compare it to the corresponding integral.

First, we find the integral:

[ \int_{1}^{\infty} \frac{x}{x^2 + 1} , dx ]

Next, we evaluate the integral. We can do this by using a trigonometric substitution or a partial fraction decomposition, both of which ultimately lead to the integral being expressed in terms of arctangent:

[ \int \frac{1}{1 + x^2} , dx = \arctan(x) + C ]

Evaluating from 1 to ( \infty ), we get:

[ \lim_{b \to \infty} \left( \arctan(b) - \arctan(1) \right) = \frac{\pi}{2} - \frac{\pi}{4} = \frac{\pi}{4} ]

Thus, the integral evaluates to ( \frac{\pi}{4} ).

According to the integral test, if the integral converges, then the series converges. Since the integral ( \int_{1}^{\infty} \frac{x}{x^2 + 1} , dx ) converges, by the integral test, the series ( \sum_{n=1}^{\infty} \frac{n}{n^2 + 1} ) also converges.

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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.

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