How do you determine if the series the converges conditionally, absolutely or diverges given #Sigma (cos(npi))/(n+1)# from #[1,oo)#?

Answer 1

#sum_{n=1}^infty{cos(n pi)}/{n+1}# converges conditionally.

Since #cos(npi)=(-1)^n#,
#sum_{n=1}^infty{cos(n pi)}/{n+1}=sum_{n=1}^infty{(-1)^n}/{n+1}#

Let us see if it is absolutely convergent.

#sum_{n=1}^inftyabs{{(-1)^n}/{n+1}}=sum_{n=1}^infty1/{n+1}=1/2+1/3+1/4+cdots#,

which is a harmonic series (divergent). So, it is NOT absolutely convergent.

Let us see if it is conditionally convergent.

Since #1/{n+1}# is decreasing and #lim_{n to infty}1/{n+1}=0#, by Alternating Series Test, we know that the series is convergent.

Hence, the series is conditionally convergent.

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Answer 2

To determine the convergence of the series ( \sum_{n=1}^{\infty} \frac{\cos(n\pi)}{n+1} ) from ( n = 1 ) to infinity, we need to analyze its absolute convergence and conditional convergence separately.

First, we examine the absolute convergence by considering the series ( \sum_{n=1}^{\infty} \left| \frac{\cos(n\pi)}{n+1} \right| ). Since ( |\cos(n\pi)| = 1 ) for all integer ( n ), the series reduces to ( \sum_{n=1}^{\infty} \frac{1}{n+1} ), which is a harmonic series. The harmonic series diverges, therefore ( \sum_{n=1}^{\infty} \left| \frac{\cos(n\pi)}{n+1} \right| ) diverges.

Next, we consider the conditional convergence by analyzing the original series ( \sum_{n=1}^{\infty} \frac{\cos(n\pi)}{n+1} ). Since ( \cos(n\pi) = (-1)^n ) alternates between -1 and 1 for odd and even values of ( n ) respectively, the series alternates in sign. By the Alternating Series Test, if ( \lim_{n \to \infty} \frac{1}{n+1} = 0 ) and ( \frac{1}{n+1} ) is decreasing, then the series converges conditionally. Both conditions are satisfied, thus the original series ( \sum_{n=1}^{\infty} \frac{\cos(n\pi)}{n+1} ) converges conditionally.

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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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