1. 11.2 Convergent or divergent series
Rita Korsunsky

2. Series vs. Sequence:
Sequence is a collection of numbers that are in one to one correspondence with the positive integers.
0.6 , 0.06 , , , …
Series is an expression that represents an infinite sum of numbers. …
kth partial sum Sk of the series an
Sk = a1 + a2 + … + ak.
s1 = 0.6
s2 = = 0.66
s3 = = 0.666
s4 = = …
Sequence of partial sums of an
S1, S2, S3, …, Sn, ….
0.6, 0.66, 0.666, , , ….
Series an converges if sequence of partial sums {Sn} converges
an diverges if {Sn} diverges. A divergent series has no sum.
0.6, 0.66, 0.666, , , ….  2/3 …

3. Example 1 series converges, sum= 1. series diverges. Given the series:
(Telescoping series)
Find S1, S2, S3, S4, S5, and S6.
Find Sn.
Converges or diverges?
Solution:
(a)
S1 = 1,
S2 = 1 + (-1) = 0,
S3 = 1 + (-1) + 1 = 1,
S4 = 0,
S5 = 1,
S6 = 0.
(b)
(c)
(c)
series converges, sum= 1.
series diverges.

4. Prove: The Harmonic Series is a divergent series.
( ) ( ) ( ) ( )
Group: S4 S8
S16
Notice that:
{Sn} diverges, so series is divergent.

5. Proof:

6. Example 3 Prove that the following series converges, and find its sum:
SOLUTION:
Geometric, r = |r| < 1, so it converges to
SOLUTION:
Geometric, |r| = 1/3 < 1, so it converges to

7. Example 4: Find all values of x for which
converges, and find the sum of the series.

8. If a series an is convergent, then
Theorem:
If a series an is convergent, then
PROOF:
n th term test:
Illustration:
Series nth term test Conclusion
Diverges by nth term test
Further investigation necessary
Further investigation necessary
Diverges by nth term test

9. Theorem Example 4 For any positive integer k, the series
either both converge or both diverge.
Example 4
Show that the following series converges:
Solution:
Delete the first two terms of the telescoping series which converges.
The series converges.

10. If an and bn are convergent series with sums A and B, respectively,
Theorem
If an and bn are convergent series with sums A and B, respectively,
Example 5
Prove that the series converges, and find its sum:
Series converges and has the sum = 10

11. Diverges Theorem Example 6
If an is a convergent series and bn is divergent,
then (an + bn) is divergent
Example 6
Determine if diverges or converges.
Geometric, |r| = 1/5 < 1, converges
Divergent harmonic
Diverges

12. End