SEQUENCES AND SERIES
Sequence – a list of terms that are in order; e.g. {1, 3, 5, 7....} or
a n =1 + 2n
∞
∑ ( 1 + 2 n)
Series – a sum of terms of an infinite sequence; e.g. 1+3+5+7+... or
To determine if series converges, you need to know the following tests:
Test
When to Use
When series has the form
Geometric Series
P -Series
Divergence Test
n=0
∞
∑ ar n − 1
n= 1
or
∞
∑ ar n
n= 0
When series has the form
∞
∑
n= 1
1
np
Conclusions
a
Converges to
diverges if
r ≥1
Converges for p > 1
diverges for p ≤1
lim a n ≠ 0
Any series for which
If
n ∞
inconclusive if
lim a n ≠ 0
n ∞
;
series diverges;
lim a n = 0
n ∞
Converges absolutely if
Ratio Test
Any series especially
those with exponentials
and/or factorials
diverges if
For series with n th powers
or greater
lim a n+1
an
n ∞
lim a n+1
< 1;
a n >1;
a n+1
inconclusive if lim
=1
n ∞ an
n ∞
Converges absolutely if
Root Test
if r < 1
1-r
diverges if
lim
n ∞
inconclusive if
n
lim
n ∞
lim
n ∞
an
n
n
>1;
an
=1
an
< 1;
To determine if a sequence converges or diverges, check the limit as
A sequence
a n converges to L if
a n = L.
n ∞
n
∞.
If limit does not exist, a sequence diverges.
Identify the advisable convergence test to use on the following series then use the test to determine whether the
series converges or diverges.
1.
2.
3.
4.
∞
∑
2k
Test:
Conclusion:
∑ n 2 e -n
Test:
Conclusion:
∑
1
Test:
Conclusion:
∑
-3n + 5
Test:
Conclusion:
k =0 e k
∞
n =1
∞
n =1 n
∞
n =0
(
2 3
3
9n + 4
3
n
(
ANSWERS:
1. Geometric series, converges
2. Ratio test, converges absolutely
3. P-series, diverges
4. Root test, converges absolutely
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