Sequences and Series Review
Two very important (distinct) mathematical objects are
sequences
and
se
ries
.
A sequence is an infinite
list
of numbers, whereas a series is an infinite
sum
. With a series
∑
∞
n
=1
a
n
, though, we can associate
two sequences:
1) its sequence of partial sums (
S
n
), where
S
n
=
∑
n
k
=1
a
k
2) its sequence of terms (
a
n
).
Our primary concern with both sequences and series is
convergence
. A se
quence (
a
n
) converges if lim
n
→∞
a
n
=
L
for some real number
L
.
A series
converges if lim
n
→∞
∑
n
k
=1
a
k
=
L
for some real number
L
; this is the same as
saying the sequence of partial sums (
S
n
) converges. If a sequence or series does
not converge, we say it diverges.
We have developed many tools for determining whether a given sequence or
series converges. The ultimate goal of all these tests is applying them to Taylor
series. We know that if the Taylor series of
f
at
a
converges for a given value
of
x
, then it converges to
f
(
x
). This is why we only test for convergence rather
than try to evaluate these series: once we have convergence, we know right away
what it converges to.
Sequence
Series
What it is
list
sum
Written
(
a
n
)
∑
∞
n
=1
a
n
a
1
, a
2
, a
3
, . . .
a
1
+
a
2
+
a
3
+
· · ·
Converges if
lim
n
→∞
a
n
=
L
lim
n
→∞
∑
n
k
=1
a
k
=
L
lim
n
→∞
S
n
=
L
Tests
Limit Laws, Substitution Law
n
th Term Test
Squeeze Law
Integral Test
Limits of Functions and Sequences
Comparison Test
Bounded and Monotone
Limit Comparison Test
L’Hˆopital’s Rule
Alternating Series Test
Ratio Test
Root Test
p
Series
Geometric Series
1
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1
Sequences
Limit Laws, Substitution Law
These laws tell us how to compute limits of sequences. If we can determine
what a sequence converges to, then trivially it converges. Recall the Substitution
Law says if
1) lim
n
→∞
a
n
=
L
2)
f
is continuous at
L
,
then lim
n
→∞
f
(
a
n
) =
f
(
L
).
Squeeze Law
The Squeeze Law is very intuitive:
if
a
n
≤
b
n
≤
c
n
and lim
n
→∞
a
n
=
lim
n
→∞
c
n
=
L
, then lim
n
→∞
b
n
=
L
also. Often we apply this with sin and
cos, bounding them between
−
1 and 1.
Limits of Functions and Sequences
This is the idea that if lim
x
→∞
f
(
x
) =
L
, then lim
n
→∞
f
(
n
) =
L
also. For
example, lim
x
→∞
2 + 1
/x
= 2, so lim
n
→∞
2 + 1
/n
= 2.
This rule essentially exists to justify using L’Hˆopital’s Rule on sequences.
Bounded and Monotone
By a theorem, any sequence (
a
n
) which is bounded (for some positive number
M
, and for all
n
,
−
M
≤
a
n
≤
M
) and monotone (for all large enough
n
, either
the
a
n
’s are increasing or decreasing) converges.
This theorem can only tell us
that
a sequence converges, not
to what
it
converges.
L’Hˆopital’s Rule
L’Hˆopital’s Rule says that if lim
n
→∞
f
(
n
) and lim
n
→∞
g
(
n
) are both 0 or
both
∞
, then
lim
n
→∞
f
(
n
)
g
(
n
)
= lim
n
→∞
f
′
(
n
)
g
′
(
n
)
.
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 Spring '09
 MAYFIELD
 Calculus, Geometric Series, Sequences And Series, Mathematical Series

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