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Midterm

MAT136H1 Study Guide - Midterm Guide: Integral Test For Convergence, Improper Integral


Department
Mathematics
Course Code
MAT136H1
Professor
all
Study Guide
Midterm

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11.3 Infinite Sequences & Series
Integral Test & Sum Estimates
Question #2 (Medium): Integral Test for Convergent Series
Strategy
For function is continuous, positive, and decreasing over , let  
The series
 is convergent if its improper integral 
is convergent
If the improper integral 
is divergent, then the series
 is also divergent
Sample Question
Use the integral test to determine if the series is convergent or divergent.


Solution
The series function  
is continuous, positive, and decreasing over , so the
integral test can apply.
Then set up the improper integral: 
   
  





 



Therefore, since the improper integral converges, the series also converges according to the integral test.
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