MAT 1236Calculus III

Section 11.2Series Part II

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HW… WebAssign 11.2 Part II Quiz: 11.2 II

Part II Introduce Geometric Series, Harmonic

Series Test for Divergence

Standard Series #1 Geometric Series (G.S.)

= first term, = common ratio

2

1

1 araraarn

n

Standard Series #1 Geometric Series (G.S.)

= first term, = common ratio

2

1

1 araraarn

n

If |r|<1, then

If |r|1, then is divergent

raar

n

n

11

1

1

1

n

nar

Proof:If , then

1

n

nk

S a na

lim limnn nS na

2

1

1 araraarn

n

Proof:

1 1

1 2

11 1

1 2 2

(1 ) 1

1

1

n nk k

nk k

n n nk k k n

nk k k

nn

n

n

S ar a ar

r S ar ar ar ar

r S a r

a rS

r

1r

Proof:

1 1

1 2

11 1

1 2 2

(1 ) 1

1

1

n nk k

nk k

n n nk k k n

nk k k

nn

n

n

S ar a ar

r S ar ar ar ar

r S a r

a rS

r

1r

Proof:

1 1

1 2

11 1

1 2 2

(1 ) 1

1

1

n nk k

nk k

n n nk k k n

nk k k

nn

n

n

S ar a ar

r S ar ar ar ar

r S a r

a rS

r

1r

Proof:

1If 1, lim lim

1

1If r 1, lim lim

1

n

nn n

n

nn n

a rr S

r

a rS

r

1r

including the case 1r

PPFTNE State and prove the convergence of the

geometric series.

Example 3

11

12kk

2

1

1 araraarn

n

Example 3Please pay attention to the important

details of the solutions Identify the series as G.S. with the

parameters a, and r From the absolute value of r, conclude

that the series is convergent or divergent (Determine the sum if it is required)

Example 4

32

1

1

23

23

231

23)1(

n

nn

Example 5Find the value of x for which is convergent

0 2nn

nx

Standard Series #2 Harmonic Series

41

31

2111

1n n

The harmonic series is divergent

Note: It is not intuitively obvious that the harmonic series is divergent.

Proof: (skip)

TheoremIf is convergent then

Why?

1nna 0lim

nna

TheoremIf is convergent then

1nna 0lim

nna

nLLa

nLaaaSnLaaaaS

n

nn

nnn

as 0

as )( as )(

1211

121

TheoremIf is convergent then

1nna 0lim

nna

nLLa

nLaaaSnLaaaaS

n

nn

nnn

as 0

as )( as )(

1211

121

?

TheoremIf is convergent then

1nna 0lim

nna

nLLa

nLaaaSnLaaaaS

n

nn

nnn

as 0

as )( as )(

1211

121

Test For DivergenceIf , then is divergent

1nna0or DNE lim

nna

Example 6

1 1n nn

By the Test for Divergence,…

Example 6Important details of the solutions Show that the limit of is nonzero (or

DNE). Make sure you put down “” (or DNE) Quote the name of the test Make the conclusion

PPFTNET or F?

If , then is convergent

1nna0lim

nna

TheoremIf and are convergent series, then

i ii m i m

i i i ii m i m i m

c a c a

a b a b