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SERIES: PART 1SERIES: PART 1Infinite Geometric Series
ProgressionsProgressionsArithmetic
Geometric
Trigonometric
Harmonic
Exponential
Infinite SeriesInfinite Series
......11
1 32
nxxxxx
Finite series always produces real numbers. An infinite series is something else entirely.
Infinite SeriesInfinite SeriesAdd terms one at a time from
beginning to view the pattern of partial sums
Find partial sum of◦1st, 2nd, 3rd …. nth term
Is there a pattern?
i.e. in assigning a meaning to this progression
...16
1
8
1
4
1
2
11
Understanding: Infinite Understanding: Infinite SeriesSeries
12
12
nnS
Momentarily, we conclude that the series converges at 2. But, can the sum of the infinite series be 2? No. Can we add the infinite series one-by-one? No. But, we can define the sum of the limit as ninfinity
Infinite Series: DefinitionInfinite Series: Definition......321 naaaa
na is the nth term of the series Partial sum of the series form a sequence
.
.
.
.
.
.
1
3213
212
11
n
kkn aS
aaaS
aaS
aS
Hence, the series converges
Converging and Diverging Converging and Diverging SeriesSeries
kS convergent
S divergent
Geometric SeriesGeometric Series
1
112 ......n
nn arararara
Where a and r are fixed real numbers and a ≠ 0. Ratio r can be negative or positive.
Term n 1 n
n arT
If | r | ≠ 1, convergence can be determined
)1(
)1(
r
raS
n
n
’ 1r
If
n
n
rrIf
rrIf
,1||
0,1||
Converge
Diverge
Geometric Series: Geometric Series: ExamplesExamples
Geometric Series: Geometric Series: ExamplesExamples1. Determine whether each series
converges or diverges. If it converges, give its sum:-
....222
)(
...8
1
4
1
2
11)(
)2
1(3)(
32
1
1
c
b
an
n
Geometric Series: Geometric Series: ExampleExample2. Find the value of the infinite
geometric series ...8
1
4
1
2
1124
Geometric Series: Geometric Series: ExamplesExamples
3. Find the sum of the infinite geometric series
4. Given the second term of a geometric sequence is ½ and the 4th term is 1/8. Find the sum to infinity.
5. Find the condition of x so that
0 2
)1(
rr
rxconverges.
Evaluate this expression when x = 1.5
....3
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3
5
3
5
3
5
3
513
11
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9
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7
4
53
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