series: part 1 infinite geometric series. progressions arithmetic geometric trigonometric harmonic...

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

5

3

5

3

5

3

5

3

513

11

10

9

7

7

4

53