How do you use the integral test to determine if #Sigma arctann/(n^2+1)# from #[1,oo)# is convergent or divergent?

Answer 1
The integral test for convergence states that for the series #sum_(n=N)^oof(n)#, if #int_N^oof(x)dx# converges to any value then the series converges as well. This only applies when #f(n)# is positive and increasing for #n>=N#, which holds for #arctann//(n^2+1)#.
So, let's test the integral for #sum_(n=1)^ooarctann/(n^2+1)#:
#int_1^ooarctanx/(x^2+1)dx=lim_(brarroo)int_1^b arctanx/(x^2+1)dx#
Let #u=arctanx#, which implies that #du=1/(x^2+1)dx#. Don't forget to change the bounds of the integral:
#=lim_(brarroo)int_(pi//4)^arctanbucolor(white).du=lim_(brarroo)[1/2u^2]_(pi//4)^arctanb#
#=lim_(brarroo)1/2arctan^2b-1/2(pi/4)^2#
Note that #lim_(brarroo)arctanb=pi//2#:
#=1/2((pi/2)^2-(pi/4)^2)=(3pi^2)/32#
Since this integral converges (the value #3pi^2//32# is irrelevant), the series #sum_(n=1)^ooarctann/(n^2+1)# converges as well.
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Answer 2

To use the integral test to determine if ( \sum \frac{\arctan(n)}{n^2 + 1} ) from ( n = 1 ) to ( \infty ) is convergent or divergent, we need to compare the given series with the corresponding integral.

  1. Define the function ( f(x) = \frac{\arctan(x)}{x^2 + 1} ).

  2. Check if ( f(x) ) is continuous, positive, and decreasing for ( x \geq 1 ).

  3. If ( f(x) ) satisfies the conditions, then integrate ( f(x) ) from ( 1 ) to ( \infty ). If the integral converges, then the series also converges. If the integral diverges, then the series also diverges.

  4. Calculate the integral ( \int_1^\infty \frac{\arctan(x)}{x^2 + 1} , dx ).

  5. Determine whether the integral converges or diverges.

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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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