Integral Test
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Suppose f is a continuous, positive, decreasing function on
and let an = f(n).
Then the series is convergent if and only if the improper integral is convergent.
Integral Test - Basic Idea
Using the Integral Test for Series
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More Lessons for Calculus
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What is the The Integral Test?
The Integral Test enables us to determine whether a series is convergent or divergent without explicitly finding its sum.Suppose f is a continuous, positive, decreasing function on
and let an = f(n).Then the series
is convergent if and only if the improper integral 
is convergent.
If is convergent then is convergent.
If is divergent then is divergent.
Example:
Test the series 
for convergence or divergence.
Solution:
The function 
is continuous, positive, decreasing function on [1,∞) so we use the Integral Test:
Since 
is a convergent integral and so, by the Integral test, the series is convergent.
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