CALCULUS CONVERGENCE AND DIVERGENCE

CALCULUS CONVERGENCE AND DIVERGENCE

TEST NAME

SERIES

CONVERGES DIVERGES

ADDITIONAL INFO

nth TERM TEST

X 1

an

=1 n

if lim 6= 0 n!1 an

One should perform this test first for divergence.

GEOMETRIC SERIES TEST

X 1

n1

anr

n=1

P-SERIES TEST

X 1 1

p

=1 n

n

if 1

1

if | | 1 r

if 1 p

If convergent, converges

to

sn

=

a 1r

Can be used for comparison tests.

INTEGRAL TEST

X 1 ()

fx

n=1

DIRECT COMPARISON TEST

X 1

an

=1 n

if

R1

1

() fx

?

dx

converges.

if

R1

1

() fx

?

dx

diverges.

( ) has to be continufx ous, positive, decreasing on [1 1).

,

if 0 , and P1 an bn

converges. bn

n=1

if 0 , and P1 bn an

diverges. bn

n=1

For convergence, find a larger convergent series. For divergence, find a smaller divergent series.

LIMIT COMPARISON TEST

X 1

an

=1 n

P1

if

converges,

bn

n=1

and lim an 0. !1 > n bn

P1

if

diverges,

bn

n=1

and lim an 0. !1 > n bn

If necessary, apply L'Hospital's Rule. Inconclusive if lim an = 0 or 1. n!1 bn

ALTERNATING SERIES TEST

X 1 ( 1)n+1 an

n=1

RATIO TEST ROOT TEST

X 1 an

n=1

X 1 an

n=1

if

an+1

, an

and

lim = 0.

if lim 6= 0. n!1 an

!1 an

n

To prove convergence prove that the sequence is decreasing and its limit is zero.

if lim an+1 1. if lim an+1 1. The test fails if

!1

<

n

an

!1

>

n

an

lim an+1 = 1.

!1

n

an

p

p

if lim n | | 1. !1 an <

n

if lim n | | 1. !1 an >

n

The ptest fails lim n | | = 1.

if

!1 an

n

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xxx

@csusm_stemcenter

Tel: North: 760-750-4101 South: 760-750-7324

CALCULUS CONVERGENCE AND DIVERGENCE

DEFINITION OF CONVERGENCE AND DIVERGENCE

P1

An infinite series

an

=

a1

+ a2

+ a3

+ ...

is

convergent

if

the

sequence

{} sn

of

partial

sums,

where

=1

n

Pn

each partial sum is denoted as = sn

an

=

a1

+

a2

+

.

.

.

+

, an

is

convergent.

=1

n

If the sequence { } is divergent, then the series is called divergent.

sn

ABSOLUTELY CONVERGENT

CONDITIONALLY CONVERGENT

P A series an is called absolutePly convergent if the series of the absolute values | | is

an convergent.

P

A series

is called conditionally convergent

an

if it is convergent but not absolutely convergent.

P1

P1

? =?

c an c an

n=1

n=1

P1

P1

P1

( + )=

+

an bn

an

bn

n=1

n=1

n=1

P1 ( an

n=1

P1

)=

bn

an

n=1

P1

bn

n=1

POWER SERIES

A power

series is a series of the form

P1

n

cnx

=

c0

+

c1x

+

2

c2x

+

3

c3x

+

...

where

x

is

a

variable

and

the

's are called the coe

cients

of

n=0

the series.

cn

P1

A series of the form

(

cn x

)n a

=

c0

+

c1

( x

) a

+

c2

( x

)2 + is called a power series in (x a ...

a)

or a power series cne=n0tered at a or a power series about a.

P1

For a given power series

( )n there are only three possibilities:

cn x a

=0

n

(i) The series converges only when x = a.

(ii) The series converges for all x.

(iii) There is a positive number R such that the series converges if |x a| < R and diverges if |x a| > R.

1 =1+ + 2+ 3+

X 1

=

n || 1

1

x x x ...

x x<

x

=0

n

P1

If the power series

( )n has radius of convergence 0, then the function defined by

cn x a

R>

=0

P1

n

( )=

( )n is dierentiable on the interval (

+ ) and

fx

cn x a

a R, a R

n=0

(i) 0( ) = P1 (

)n 1.

fx

ncn x a

n=0

(ii)

R

P1

( )= +

(x

fx C

cn

. )n+1

a +1

=0

n

n

csusm.edu/stemsc

xxx

@csusm_stemcenter

Tel: North: 760-750-4101 South: 760-750-7324

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