Math 253 Cheat Sheet - Sequences And Series
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1Math 253 Cheat Sheet - Sequences and Series
Test
Notes
n
1 r
n+1
i
Geometric Series (finite)
r
=
(r = 1)
1 r
i=0
n
n(n+1)
Triangular Numbers
i =
2
i=0
1
|r| < 1
∞
n
1 r
Geometric Series (infinite)
r
=
|r| ≥ 1 Divergent
n=0
∞
Telescoping Series
(a
a
) = a
a
n
n+1
0
∞
n=0
p ≤ 1 Divergent
∞
1
p-Series
=
n
p
p > 1 Convergent
n=0
0
N o Inf o.
Test for Divergence
lim
a
=
n
other Divergent
x→∞
(1) b
< b
∀n
∞
n+1
n
n
Alternating Series Test
( 1)
b
where b
> 0 ∀ n converges if
n
n
(2)
lim
b
= 0
n
n=0
n→∞
Comparison Test
Terms Get Smaller
Terms Get Bigger
(0 ≤ a
≤ b
)
(0 ≤ b
≤ a
)
∞
∞
n
n
n
n
(Compare
a
to a known series
b
.)
n
n
Known Conv. Series
Convergent
No Info.
n=0
n=0
Known Div. Series
No Info.
Divergent
Limit Comparison Test
If
a
, b
≥ 0∀n?
n
n
a
∞
∞
and
lim
= c > 0?
n
(Compare
a
to a known series
b
.)
b
n→∞
n
n
n
∞
∞
n=0
n=0
T hen
a
converges ⇐⇒
b
converges
n
n
n=0
n=0
a
lim
= L < 1 Absolutely Convergent
n+1
a
n→∞
n
a
Ratio Test
lim
= L > 1 Divergent
n+1
a
n→∞
n
a
lim
= L = 1 T estF ails
n+1
a
n→∞
n
lim
|a
| = L < 1 Absolutely Convergent
n
n
n→∞
lim
|a
| = L < 1 Divergent
Root Test
n
n
n→∞
lim
|a
| = L = 1 T estF ails
n
n
n→∞
If
a
= f (n) is continuous, postive, and decreasing on (1, ∞)
n
Integral Test
∞
∞
T hen
f (x)dx converges ⇐⇒
a
converges
n
1
n=0
2
n
note: a little while doesn’t matter: e
/n.
Pascal’s Triangle:
1
Taylor Series
1
1
∞
(n)
(0)
f
n
f (x) =
x
(McClaurin)
1
2
1
n!
n=0
1
3
3
1
∞
(n)
(c)
f
n
f (x) =
(x
c)
1
4
6
4
1
n!
n=0
n
∞
n
d
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