STEC

1. e.9.,
t, 5,7, ...,
t'
,Ir,
A sequence is a
nth term in. a sequence
'For examPle, if un
By
A
ls
a limit I then rrye r{rite
sequence,
e,g,; 7;
I
: l, -1,
A sbquence which
'.
3. l; -t, t, -1, ;., "

2. H"t:
H
L:
t:
,
r,
l
l'
I
t,
:
i,r .,
t.
;r'
l, :
1'.t20
Which bf.rhe
lnllnite
20.9
n+ PRO
Ans. Convergeiit :, ;
':l ' il:,
Anrl Divergent ,l
l- Th
.(
{i
2. If
REMEMBER THE20.6
(vD
0)
: (i,
(rv)
=.Ql
-Errl
.Seh
Sh.t
I*q(
&r
.0 for;all val.ues of .r
,lgglnl}r/' = -
'
limrx'=0if,r<ln+6
log a or' ./ ,
.i
,I
r 'l
.I
, ,, J
serii'i
I
,
Solt
': ,. H(
Er
calledls
a
)

3. t:
,il
,,]
l
ri
'l
ili
"l. :i.
rl
!l
i
i
-'ll:':
ieries
Hence, S, does not tend
Solution. Here,
.-
.:. x rr.
s,

4. L,
.1'.|22
I
r-
Hence,
(D) When r let r

5. 1f
Hence,,the given series is convergent when P > 1.
Case2:P=f
When p,= 1, the given seiies becomes
1
+-:.
1-
-+ t6
On adding (l), (2), (3) and (4), we get
given se-ries is
L=k
r1.P or
Proof. By definition of
both
posltivo
>m.,
,tli
:
i
l:

6. ..{i$
0n
Case
I
rf.'
r trar.
.-k<
1'
+
!.

7. l.
${.j:,,;
' l:i., i.
,16
and non-zero,
;. Iun and Ivr, conveiie or
.'. Iy,
I
diverge together
..l
since f, v, , = )-a is of
. ..,. .n2.
1
2
: tl*r;.'"".r
1128 lnrirtite Series
)
ilo =
+l+
l+
Let us compare I zn wilh I Y, , where
-1- :':, i'
un
yn
I
l+
I
,.:,,
I
-n
++
1
I"€t
Iv" converg€ or
ylknPthe form withp+2>1-

8. nfirfite
' ' (M.9:U:.2ooo)
Here, we have ralt:i i .
J
..
r.'i,r.l
.r:.:,1r. fj
",,. l'.. .: 2._+_+
l+2-t l+Z-2
5
___+...
l+2''
He;e ilr=
Let us compare
'Examplc 12. Exanine the
Solution. Her"e, we have
r'ilr
I".,
= lim
}'+4.
uilx.iJl ,,
Arr
*.,r- ;l
r-t/t*;
n
I
?r = _T:
n2,
I
n
lnfinite Series
Here ,.
1179
Ans.'
i form f, a
,r
.f,.ii ,.'i
(M.D.U.'2oAJ)
Solution.
',. ; .
!$.1 .
4o=
l+
: ', '-,

9. EXERCISE 20.5
Examinc lhe converBence or divergencc of the follbwing series: ,
,- , * 3a * 1.* * q.+ *.... Ans. convergent
" -. 2'4 3 42 4 43
1.2 7.2.3 7.2.3.4'
...@ Ans. convergent2' 1+
13
*
r33
*
l-:.s.2
lnfinite Series
.1
D'
$ 2n3 +,5
*r4" t
iAne.Convergent
t,i..:
_ril il, ' ' . -,,r i.
1231
-+-+-+...@
-' 1.2 3.4 5.6
111/ L+-+-*.........@
'' r,2.3 2.3.4 3.4.5
1 131,
Ans.,Diverginti' I 'r'
Ans. Convergent .(M.D. IJniversity' Dec' 2004)
t:
'. . Ans, Convergent
'ti
Ans, ConveJgent
.,
Ans. Convergent'
.+
22
ol,[.
"afiE&
,5.
6.
:,
I
= ,,('
+
I
Jr+{r+1
.an
3t
+
Divergent
> a, convergenl; il x 3 a, DivergentAns. If r
9.
Ans. Cunvergent
R]TTIO TEST
6_',
12. Z r/{n' * t; - n ', Ans Divergent'
. A!rs. lo"pyergent
r ',,r::",, '
.. l:r r
iaTl
Ans.
2n +
T;14. Ir.l
15. 16. L;n,.1r
(l +r
Strtcmcnl. Il2 ur ls a PositUe lerm thcnel
(t) thc seiles ls
8.
11.
13.
10. itr.l 4'+no
+J;L-n2 +l,.1
Casc
By'
I
I
I
j
i
,
i
I
j
Convergent
=k
,,:, {
'.::
t
.+..., @
ll2
t *-,'l .
i

10. limiq
n
ofBy
)
Coruider the
t+o
Thus, froqr (1) and (2)
Hence, wheri
Thus, ritiq

11. Whictr is finite and
,.i,. ' .,:
series whose n'h term is
;2n
we have un=
-,n'
BY D'Aiembertrs Ratio Tegt
+l n9 2
rl
lr
Ur+l
4o
I
:,, t+
(4+ 1++