Explanation:- counter example: in Lan) = = (-1) :nGN sequencence of real number which is not convergent it is oscillatory. oscillate finitely be) Proofin let two sequence g so given 430 I, and I be the points of Land and [bn] respectively, J no G12 sit we have 2-₁₁ -1₁ -1, 1,-1,---- y 2-1,011 2 Nyno cans - dil 22 and | 2bn3 - 12/2€. 2 Lan) Hơn 3) kith 8 our claim 1.8. [an] tibn) cyf to i+dz ✓ N», no. we will use triangle inequality V equality. stan an-dil 올 가을 2 Bum of two 2 & ... kandbon) I LE Hence Convergent sequence is convergent. + 1 barda]
Explanation:- counter example: in Lan) = = (-1) :nGN sequencence of real number which is not convergent it is oscillatory. oscillate finitely be) Proofin let two sequence g so given 430 I, and I be the points of Land and [bn] respectively, J no G12 sit we have 2-₁₁ -1₁ -1, 1,-1,---- y 2-1,011 2 Nyno cans - dil 22 and | 2bn3 - 12/2€. 2 Lan) Hơn 3) kith 8 our claim 1.8. [an] tibn) cyf to i+dz ✓ N», no. we will use triangle inequality V equality. stan an-dil 올 가을 2 Bum of two 2 & ... kandbon) I LE Hence Convergent sequence is convergent. + 1 barda]
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Explanation:
counter example in
Lan) = (-1)
9.)
= 2-1, 41,-1,1,-1,-- y
:nGN
sequencence of real number which is not
it is oscillatory.
convergent it is
oscillate
finitely
bu) Proof: -
let
two sequence
g 30,
I, and Is
V
given Exo
Nyno
be the points of
Land and Lbn> respectively.
I no G Z
sif
1
our claim rs.
we have
Ny ho.
Lan) - do I Le and
2
[an] tibn) cyf to dit da
we will use triangle inea
inequality
1203-10 12€.
12/24
equality.
| Kean) telon)) - (Wit (₂) | = | an-dil + | b₁-d₂]
222 +2
Hence
... Kanton) | LE
Convergent sequence is convergent.
Sum of two](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6be4e98e-3d80-47d6-a8a2-7cbe1f714c88%2Fb52cfd8f-7af5-494d-9faf-70c613702cc4%2Fn67puhj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Explanation:
counter example in
Lan) = (-1)
9.)
= 2-1, 41,-1,1,-1,-- y
:nGN
sequencence of real number which is not
it is oscillatory.
convergent it is
oscillate
finitely
bu) Proof: -
let
two sequence
g 30,
I, and Is
V
given Exo
Nyno
be the points of
Land and Lbn> respectively.
I no G Z
sif
1
our claim rs.
we have
Ny ho.
Lan) - do I Le and
2
[an] tibn) cyf to dit da
we will use triangle inea
inequality
1203-10 12€.
12/24
equality.
| Kean) telon)) - (Wit (₂) | = | an-dil + | b₁-d₂]
222 +2
Hence
... Kanton) | LE
Convergent sequence is convergent.
Sum of two
Transcribed Image Text:Notes-
Definition:- gt sean,
A sequence Lan) is convergent ift
Such that for ESO
for
that.
every
170-6124
Cauchy sequences -
А seem
(an)
iff for
such that,
loe
have
Prooft- (C.): -
let
for some
such that
VE)0
nGN.
have 1xm- and LE.
there
I there in NGN
with n>N.
Land
is called Cauchy sequence
there is NGA
for every minGN. with min A,
K₁, K₂ > max
gronvergent to x, we
41,4230
, 3 n, 1₂
GA
VnKishi and K₂) ₂
1xK₁ -α1 24,
and 19*2*1 242
80
both will hold for all
is dєк
such
(n₁,1₂)
E = max [E₁, (₂) then
OK L ket).
10421-2-(2x₂-1) | LE
15x₁-xx²₂128
Proved
Hence all convergent sequence is cauchy,
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