Infinite Series Practice Problem 2436 

\(\displaystyle{\sum_{n=1}^{\infty}{\frac{n~\cos(n\pi)}{2n1}}}\) 
Determine the convergence or divergence of the series. 
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Choosing Which Tests To Try 

Rejecting the Obvious 

1. This is not a telescoping series or one of the special series.
2. We can't use the integral test because there are some negative terms and the integral test requires all positive terms.
What We Are Left With 

3. If we look closely, the term \(\cos(n\pi) = (1)^n\), so we have alternating series test. If the alternating series test does not work, we can move on to some others, but we want to try the divergence test and the alternating series test first.
Note  Once we determine that we have an alternating series, we immediately try the alternating series test, since most of the time, it will work. If it doesn't for some reason, we can come back and consider other series tests.
Applying The Tests 

\(\displaystyle{ a_n = \frac{n~\cos(n\pi)}{2n1} }\)
\(\displaystyle{
\lim_{n \to \infty}{ \frac{n~\cos(n\pi)}{2n1} } = \lim_{n \to \infty}{ \frac{\cos(n\pi)}{21/n} }
}\)
In this limit, we know that \(\displaystyle{ \lim_{n \to \infty}{1/n} = 0 }\), leaving \(2\) in the denominator. In the numerator, \(\cos(n\pi)\) oscillates between \(1\) and \(1\). So we are left with the series oscillating between \(1/2\) and \(1/2\) and therefore, the limit
\(\displaystyle{
\lim_{n \to \infty}{ \frac{n~\cos(n\pi)}{2n1} } \neq 0
}\)
So, by the divergence test, the series diverges.
Conclusion
The series \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n~\cos(n\pi)}{2n1} } }\) diverges by the divergence test.
If you look closely, you will notice that \(\cos(n\pi)\) is just a fancy way of writing \((1)^{n}\). This is an alternating series which can be written
\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(1)^{n} n}{2n1} } }\)
Let's try the alternating series test with \(\displaystyle{ a_n = \frac{n}{2n1} }\)
Condition 1: We need to find the limit of \(\displaystyle{ \lim_{n \to \infty}{a_n} }\).
\(\displaystyle{ \lim_{n \to \infty}{a_n} = \lim_{n \to \infty}{ \frac{n}{2n1} \frac{1/n}{1/n} } =
\lim_{n \to \infty}{ \frac{1}{21/n} } = \frac{1}{2} \neq 0
}\)
Since the limit is not zero, condition 1 does not hold, so the alternating series test is inconclusive.
Test Summary List and Answer 

test/series  works?  notes  


yes 
best test  
no 

not a pseries  
no 

not a geometric series  
no 

condition 1 does not hold  
no 

not a telescoping series  


 


 


 
no 

integral test requires positive terms  



Final Answer 

The series \(\displaystyle{\sum_{n=1}^{\infty}{\frac{n~\cos(n\pi)}{2n1}}}\) diverges.
Notes 

Once we noticed that we had an alternating series, we started with that; when condition 1 didn't hold, we knew that the divergence test would tell us that the series diverges.
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