Infinite Series Practice Problem 2442 

\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^{3/2}+1} } }\) 
Determine the convergence or divergence of the series. 
Recommended Books on Amazon (affiliate links)  

Choosing Which Tests To Try 

Rejecting the Obvious 

1. We can tell right away that this is not an alternating series, telescoping series or one of the other special series. So we will not be able to determine what the series converges to.
2. We can tell that the limit of \(a_n\) is zero. So the divergence test will not help us.
3. The integral is not easily evaluated, so we will come back to it if none of the other tests work.
What We Are Left With 

4. We are left with the ratio test, the comparison tests and the root test.
Applying The Tests 

Test Summary List and Answer 

test/series  works?  notes  

no 

limit is zero, so test is inconclusive  
no 

not a pseries  
no 

not a geometric series  
no 

not an alternating series  
no 

not a telescoping series  
no 

inconclusive  

yes 
 

yes 
 
no 

too complex  
no 

inconclusive 
Final Answer 

The series\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^{3/2}+1} } }\) converges.
Really UNDERSTAND Calculus
all infinite series topics 
L'Hopitals Rule 
To bookmark this page and practice problems, log in to your account or set up a free account.
Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.
 

