3.4 Conditional and Absolute Convergence
a series, an is said
to converge absolutely,if the series, an converges
if I,An converges but Iland diverges we say,an is conditionally convergent
Theorem:
17, lan) converges, ne,an also converges
example: E,(-1"
the
alternating harmonic series converges by AST
test
the harmonic series diverges by the
integral
.....(-1)"- A converges conditionally
example:E, El)" Un
il(-1)" ) 2,Une converges by integral
-
=
test
. Fi)r-"n= converges absolutely ...
converges
example:Crandom signs)
imagine flipping a coin infinitely many times:
Seton= +1 if the nth flip
comes up heads and On= -1
if the nth flip comes up tails.
The series .C-Donn is a
not
general alternating series
since we know , K-Don'n 'no converges, (-1)on'n converges absolutely.
=
-
a series, an is said
to converge absolutely,if the series, an converges
if I,An converges but Iland diverges we say,an is conditionally convergent
Theorem:
17, lan) converges, ne,an also converges
example: E,(-1"
the
alternating harmonic series converges by AST
test
the harmonic series diverges by the
integral
.....(-1)"- A converges conditionally
example:E, El)" Un
il(-1)" ) 2,Une converges by integral
-
=
test
. Fi)r-"n= converges absolutely ...
converges
example:Crandom signs)
imagine flipping a coin infinitely many times:
Seton= +1 if the nth flip
comes up heads and On= -1
if the nth flip comes up tails.
The series .C-Donn is a
not
general alternating series
since we know , K-Don'n 'no converges, (-1)on'n converges absolutely.
=
-
