How do you determine if the series the converges conditionally, absolutely or diverges given #sum_(n=1)^oo (1)^(n+1)arctan(n)#?
The series diverges.
The Alternating Series Test
And if the series doesn't converge conditionally then it doesn't converge absolutely.
By signing up, you agree to our Terms of Service and Privacy Policy
To determine if the series ( \sum_{n=1}^{\infty} (1)^{n+1}\arctan(n) ) converges conditionally, absolutely, or diverges, you would typically follow these steps:

Check if the series is alternating. In this case, the series ( \sum_{n=1}^{\infty} (1)^{n+1}\arctan(n) ) is alternating because it alternates in sign.

Check if the series satisfies the conditions for the Alternating Series Test, which states that if the terms of an alternating series decrease in absolute value and approach zero as ( n ) increases, then the series converges.

Determine if the series ( \sum_{n=1}^{\infty} \arctan(n) ) converges or diverges. This will help determine if the original series converges absolutely or not.

If the series ( \sum_{n=1}^{\infty} \arctan(n) ) converges, then the original series ( \sum_{n=1}^{\infty} (1)^{n+1}\arctan(n) ) converges absolutely.

If the series ( \sum_{n=1}^{\infty} \arctan(n) ) diverges, but ( \lim_{n \to \infty} \arctan(n) = 0 ), then apply the Alternating Series Test to the original series to determine if it converges conditionally.

If the series ( \sum_{n=1}^{\infty} (1)^{n+1}\arctan(n) ) satisfies the conditions of the Alternating Series Test, then it converges conditionally.
In summary:
 Check if the series is alternating.
 Check if the terms decrease in absolute value and approach zero.
 Determine if the series ( \sum_{n=1}^{\infty} \arctan(n) ) converges.
 Apply the Alternating Series Test if necessary to determine conditional convergence.
By signing up, you agree to our Terms of Service and Privacy Policy
When evaluating a onesided 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 onesided 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 onesided 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 onesided 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 test the series #Sigma 1/sqrt(n(n+1))# from n is #[1,oo)# for convergence?
 How do you test the series #Sigma 1/(n!)# from n is #[0,oo)# for convergence?
 How do you test for convergence for #(sin 4n)/(4^n)# for n=1 to infinity?
 How do you test the improper integral #int x^(1/2) dx# from #[1,oo)# and evaluate if possible?
 How do you determine the convergence or divergence of #Sigma ((1)^(n))/(sqrtn)# from #[1,oo)#?
 98% accuracy study help
 Covers math, physics, chemistry, biology, and more
 Stepbystep, indepth guides
 Readily available 24/7