3.3.4 The Alternating Series Test
when the signs of successive terms alternate between + and -, this is an alterating series
·A.-Az+Ag-Ayt... -
(-))n- An is
alternating if A, A2 "As...
Theorem:
letEAn32_, be numbers that
a sequence of real obeys
An is positive for all nx,I
the sequence is monotone
decreasing
An O
·
then EC-D**An=s converges and for each N, S-SN is between 0 and (-1) Ant
N
SN El-1n- An
=
example:
we know the harmonic series indiverges butthe series, (-1)n-'t converges by the alternating
series test with An=
h
note:
An=120 for all nx1
An decreases as increases
An O
...E,(-3"- converges
/
when the signs of successive terms alternate between + and -, this is an alterating series
·A.-Az+Ag-Ayt... -
(-))n- An is
alternating if A, A2 "As...
Theorem:
letEAn32_, be numbers that
a sequence of real obeys
An is positive for all nx,I
the sequence is monotone
decreasing
An O
·
then EC-D**An=s converges and for each N, S-SN is between 0 and (-1) Ant
N
SN El-1n- An
=
example:
we know the harmonic series indiverges butthe series, (-1)n-'t converges by the alternating
series test with An=
h
note:
An=120 for all nx1
An decreases as increases
An O
...E,(-3"- converges
/
