1. 11.3a: Positive-Term Series
Integral test.
P-series.
Rita Korsunsky

2. Positive-term series are the series an such that an > 0 for every n.
If an is a positive-term series and if there exists a number M such that:
Sn = a1 + a2 + … + an < M for every n,
then the series converges and has a sum S  M.
If no such M exists, the series diverges.

3. The Integral Test:
Sometimes we need to prove the right to left statements
In using the integral test it is necessary to consider
If f(x) is not easy to integrate, a different test for convergence or divergence should be used.
When we use the Integral Test it is not necessary to start the series or the integral at For instance, in testing the series
Also, it is not necessary that be always
decreasing. Instead, could be decreasing
for larger than some number . Then
is convergent, so is convergent too.

4. Total area of other rectangles <
Exclude the first rectangle.
Total area of other rectangles <
area under the curve or
Integral
converges
Partial sums are bounded =1
 Series converges
partial sums  1.64
as n   n
Total area of all rectangles > area under the curve or
Integral
diverges
Partial sums are unbounded
 Series diverges
partial sums  
as n  

5. Let’s try! Why don’t we use inscribed rectangles
Total area of all rectangles > area under the curve or
Integral
diverges
Partial sums are unbounded
 Series diverges
partial sums  
as n  
Why don’t we use inscribed rectangles
instead of circumscribed in 2nd problem?
Let’s try! 1 2 3 4
Does it converge or diverge?
Integral
diverges
So we can not make any conclusion.
Use the circumscribed rectangles instead!

6. The series diverges by the integral test.
Example 1
Prove that the harmonic series diverges
using the integral test.
Harmonic series:
f > 0, continuous, and decreasing for x  1,
so use the integral test:
The series diverges by the integral test.

7. Converges But sum Example 2 f(x) > 0 and continuous if x  1.
Does the infinite series converge or diverge?
SOLUTION:
f(x) > 0 and continuous if x  1.
f(x) is decreasing
Converges
But sum

8. P-series (hyperharmonic series)
where p > 0
Proof
Use Integral
test.

9. ILLUSTRATION:
p-series value of p conclusion
p = 2 Converges, p=2 > 1
p = 1/2 Diverges, p=½ < 1
p = 3/2 Converges, p=3/2 > 1
p = 1/ Diverges, p=1/3 < 1