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
n 1
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 1 r
Can be used for comparison tests.
INTEGRAL TEST
X 1 ()
fx
n=1
DIRECT COMPARISON TEST
X 1
an
=1 n
if
R 1
1
() fx
?
dx
converges.
if
R 1
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
csusm.edu/stemsc
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
coecients
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
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related download
Related searches
- convergence divergence calculator
- calculus problems and answers pdf
- calculus problems and solutions pdf
- calculus derivatives problems and solutions
- calculus integration problems and solutions
- differential and integral calculus examples
- advanced calculus problems and solutions
- differential and integral calculus pdf
- ap calculus problems and solutions
- thomas and finney calculus pdf
- calculus velocity and acceleration problems
- calculus questions and answers pdf