section 8.2: infinite series. zeno’s paradox can you add infinitely many numbers ?? you can’t...
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Section 8.2: Infinite Series
Zeno’s Paradox
Can you add infinitely many numbers ??
1
2
1
4
1...
8 ? ?
You can’t actually get anywhere because you always have to cover half
the remaining distance!
You have to do half, then half that.. etc.
Informal Definition
A series is sequence added up.
A series converges
if the sequence of partial sums converges.
Formal Definition
1k
k
a
1
n
n kk
S a
Example
1
1
2
k
k
1
n
n kk
S a
1
2
1
4
1...
8
1
1
2S
4
1 1 1 1
2 4 8 16S
3
1 1 1
2 4 8S
2
1 1
2 4S
2
0
...k
k
ar a ar ar
Definition
is a geometric series.
0
k
k
ar
Theorem
Diverges if
Converges to if
| 1|r
1
a
r| 1|r
Proof:
nS
nr S
(1 ) nr S
nS
0
k
k
ar
... na ar ar 1... n nar ar ar
a
1
(1 )
na ar
r
1
a
r
limn
limn
If and only if| | 1r
1nar
0
2
3
k
k
12
13
3
0
1
2
k
k
1 1 1
2 41
8 2
11
12
1
1
2
k
k
1 1 1
2 4 8 1
112
1
This solves Zeno’s paradox!!
1
2
283
k
k
1
2
243
k
kk
0
283
k
k
2
2 243
* k
kk
2
283k
k
81
82
13
24
2
2824
383
28
k
k
32
3
283
1
12
415
k k
kk
2
420
5
k
k
20204
15
100
9
2
100
9
420 2
420
550
k
k
2
54 * 4
15
k k
kk
0
420
5
k
k
420
5
2
420
5
k
k
64
9
11
372
k
kk
1
314
2
k
k
Diverges
Telescoping Series
1
1
( 1)k k k
Use Partial Fractions1
1 1
( 1)k k k
nS 1 1
2 3
112
1 1...
1n n
11
1n
1 1
3 4
lim nnS
1
1
( 1)k k k
11lim 1
1n n