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Transcript
Section 11.2:
Series
Infinite Series of Real Numbers
1. Sequence of terms:
a1 , a2 , a3 ,
2. Sequence of partial sums:
S1 , S 2 , S3 ,,
3. Series:

a
i 1
i
n
where S n   ai
 a1  a2  a3  
i 1
th
n
Partial Sum and
th
n
Tail
n
S n  a1  a2  a3    an 1  an   ai
i 1
Rn  an 1  an  2  an 3   
n
S n  Rn   ai 
i 1


a
i  n 1

a  a
i  n 1
i
i 1
i
i
Convergence of a Series
If lim S n  S for some finite number S, then the
n
series


a converges to the limit S. Otherwise,
i 1 i
the series diverges.
The Divergence Test
• If the sequence {an} does not converge to 0,
then the series


diverges.
a
n
n 1
The Harmonic Series

1
1 1
1
 1   

2 3
n
n 1 n
Geometric Series

•
n 1
2
n 1
ar

a

ar

ar



ar


n 1
• Convergence
a
.
– If r  1, converges to
1 r
– If r  1, diverges .
Algebra with Series



 a  b    a   b
i 1
i
i
i 1

 ca
i 1
i
i

 c ai
i 1
i 1
i