11/o¢/24
Jnfiase Series
and sor
Where
Unis
the
h terr m alled t
qensral trm
145-
Oslintory
Hnl thenthe
Segenee is caid to be a
Convergent
Andif
Seunce, Converqy to l.
n--to
m n d,
n’
Squene is
thwthe
,
sad ta
be dives
, n
Divergt
Cow ergent
Infiuite Seriey
is called an
Series.
Patial Sum
T
5/,
Journ
ean
neart
Now, Con veLgene or divegence bf the sees
on he onvbqen ordingena
of the ceg uen ee Fsn? ofnt ppootial sum
’The sequenc Sn convgyfo n, anmd the
we cay that the seie
n
Converge
The
and Conver gen e to
Sm
the seies is Said to be
Ainyat
dseyo have amt
tve value
tn
seri
s sa
to e
fite osiMatorg b/u the valu o Cnts.
) foy
the
seies,
for the SeriesY
S =lt2 +
Snd
n
+n
n(ntI)
henral
roprtiey of Behanor o seics
The bchaviow. ( Conveou, diveye ordelay
of an nft nite soey wemais inafferted
the adlton or emoval o a f i e
no
(ü)The behaviour of an infnide seies
remain
un afec ted if each tesm i s
mutilid by a inite non-zo0 lowfart
(e
Londition For
or comrn a
A wets$ary Londition
Convemgd, thn
20/08/24
GEONEt2IC SERIES
a +aLt
tufimiae
4 -
ash
is ca Led a
aomt'e cries
tum a
tS commenYatia
() The Aeometie seien is tonv
0s lato
fist
T
4he Soues
n (nt )
(ht)
(nt)
im
Convegut
Test for
SUuey
divergt
o
comengnt Seiey i
casey,'we Cn in d the
sum. 1hws we Ca n-fnf
Some Case -fo See
whethet
potihve
n dverys,
tthehe
also
sUn
Convrge
or
the
qe
diverge
to
n bo-th then,
amd
vrges
abo
Un -+
Seies
ln
n then
- tVm
enies.
get
serue) s9ent
tonverg
>
if
e8
-
-t
2Vn
is
aaloo
Jdy
div
es
s,thn
is n
t
1t
obeao
Un then Soles
0nd integen htve
a be7Un
t
tnym
n sereey
em
VC
(Opaac0n
t 2 - n-)
Cowirg
1.2-2 h-2) f'msy
(4)
Pa
J
S
)
(noN
46AZ
=+)
+
(o
(as
(P<)
divrgot
. t hey efme, bg com paision test seues
seues.
divogent
divngent seue.
S Un is also a
The
Convgen t:
beeawse
is a G. S oi-h cRt
anvegunt
+ 3
53
(t
R
25
a
55
+
a=c.R=
5
’ onvegent
+
42
P>t)
Convegent
13
n n
)
Al.bnls Ratio Te t
2ln be
cincluion can be draon t l l
D'A Lemab nts
Ratio
Test
2un be
term seiei
convog
And no
) 2>1
concluin can be draon t 1 .
whet
te
2
And
)-1
fnd
whethe
the s s i X +
ynt
divgen
is con vengnt
-+
+
is +ve
whe
im
nt |
n
convegt
iveent
Can ae drawn.
ivag
Mn
Vn
fr.
-eefee the
un
åicuss the convog
3
+
4
the seiey
(e>o)
1 by
(nt)e
(nt)
,n(n+)
)
P
im
9
()
2
( nt)
(nt)
n(nt
nt Soiey
Con vergent
(1H
)
Ranbe's ? est
#re
of
soues of
u, is a soies
ut,
A
series
term
a
n
-)
ntt
n
Convoyey it,
diveneib
no deeusi on c an ae
ran
1
+1
l(
(t+wz)
+u)
X
(1+u
-+)
X(n)
(nt)
2)'4dn+
(nt2n)
(
Wm
mtant2
(2
tl
,n+
hatConclude
can
weteut, )faabe's
2
4X=
.
lim
fuRsabe'
ess
shall
wnda
Now,
(onlusion
bo
Cam
utt,c
wyr
)tzu
(t-)
(
(
vmn
6ausu Tet
-tve to1n3'
be
t
boundod
)
value
Aiverrg
Conrerqes
dirur
lest
seiey
Seuh that
im
f tveerm
(unya
Convege
C a i t u can
be d a h
()
uri
(n
2
34
nt|
nt|
w
donerget ories
(Zn+1)2n
(2nt2+1)(2n+2)
. 2n(2n+1)
(2n+3) (2+ 2)
Chwwrgent it, <|
Aivegint ,*>1
cahelson (an be dran.
(
an+2n
4