Proof by induction: Base case: trynna see if the sequence is increasing. n=1 Xn € Xn+1 by definition X₁ = × 1+1 VEN x₁ = X2 √ ≤ 2 ✓ Induction Steps: Let n = k Э KEIN we know to show that and xx = Xxxl we are trying Xx+1 £ xk+2 Therefore, +2 (πx) = (xx+1)+3 √2+x=12+ X1 Xx+1 = Xk+2 Thus we can conclude that the sequence {X} Sequence. by the monotone definition is an increasing Now problem we know we need to show if it 15 bounded in the X ≤ 2 and given √2 = Proof by Induction: Base case: we know √2 ≤ X ≤ 2, UnEN √2 ± X₁ = 2 √2 ± √2 ± 2 ✓ Induction Steps: Let n = k B KEN ≤ and we want to we know √2 = Xx 2 Show that So. Xx+ 1 ≤ 2. +2 (√2)+2(x)(2) +2 √2+82 €12+ x =√4 2≤ Укн ≤2 Therefore sequence {Xx} In conclusion n In base on this we know that the is bounded. Show that the sequence {K^} is Therefore has the definition we d bounded In conclusion increasing and ' we show that the sequence 9x] 15 and bounded. Therefore by the definition of monotone convergence {v} is converges. thereom this sequence Now we need to find the limit: * Let X" be the limit lum (date) = √2+ km (x) 1im (xn+1) 7*2 +#2 - χ 2 2+ 2 + x* x-2=0 (x*+1) (x* -2) X=-1 = 0 x = 2 ** The limit is 2 2. Let the sequence (xn) be recursively defined by x₁ = √√2 and Xn+1 = √2+xn, n≥1. Show that (xn) converges and evaluate its limit.
Proof by induction: Base case: trynna see if the sequence is increasing. n=1 Xn € Xn+1 by definition X₁ = × 1+1 VEN x₁ = X2 √ ≤ 2 ✓ Induction Steps: Let n = k Э KEIN we know to show that and xx = Xxxl we are trying Xx+1 £ xk+2 Therefore, +2 (πx) = (xx+1)+3 √2+x=12+ X1 Xx+1 = Xk+2 Thus we can conclude that the sequence {X} Sequence. by the monotone definition is an increasing Now problem we know we need to show if it 15 bounded in the X ≤ 2 and given √2 = Proof by Induction: Base case: we know √2 ≤ X ≤ 2, UnEN √2 ± X₁ = 2 √2 ± √2 ± 2 ✓ Induction Steps: Let n = k B KEN ≤ and we want to we know √2 = Xx 2 Show that So. Xx+ 1 ≤ 2. +2 (√2)+2(x)(2) +2 √2+82 €12+ x =√4 2≤ Укн ≤2 Therefore sequence {Xx} In conclusion n In base on this we know that the is bounded. Show that the sequence {K^} is Therefore has the definition we d bounded In conclusion increasing and ' we show that the sequence 9x] 15 and bounded. Therefore by the definition of monotone convergence {v} is converges. thereom this sequence Now we need to find the limit: * Let X" be the limit lum (date) = √2+ km (x) 1im (xn+1) 7*2 +#2 - χ 2 2+ 2 + x* x-2=0 (x*+1) (x* -2) X=-1 = 0 x = 2 ** The limit is 2 2. Let the sequence (xn) be recursively defined by x₁ = √√2 and Xn+1 = √2+xn, n≥1. Show that (xn) converges and evaluate its limit.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Im trying to solve this problem based on what i known but i feel like my proof to check to see if the sequence is bounded is incorrect so can someone please take a look see if i did it right and also the whole proof for the problem please? Thank you very much.
![Proof by induction:
Base case:
trynna see if the sequence is increasing.
n=1
Xn €
Xn+1 by definition
X₁ = × 1+1
VEN
x₁ = X2
√ ≤ 2 ✓
Induction Steps:
Let n = k
Э
KEIN
we know
to show that
and
xx = Xxxl
we are trying
Xx+1
£
xk+2
Therefore,
+2 (πx) = (xx+1)+3
√2+x=12+ X1
Xx+1 =
Xk+2
Thus we can conclude that the sequence {X}
Sequence. by the
monotone definition
is an
increasing
Now
problem we know
we
need
to
show
if it 15
bounded
in
the
X ≤ 2
and given
√2 =
Proof by Induction:
Base case:
we know √2 ≤ X ≤ 2, UnEN
√2 ± X₁ = 2
√2 ± √2 ± 2
✓
Induction Steps: Let
n = k
B KEN
≤
and we
want to
we know √2 = Xx 2
Show that
So.
Xx+ 1 ≤ 2.
+2 (√2)+2(x)(2) +2
√2+82 €12+ x =√4
2≤ Укн ≤2
Therefore
sequence {Xx}
In conclusion
n In
base
on
this
we
know that the
is bounded.
Show that the sequence {K^} is
Therefore has the definition
we
d
bounded
In conclusion
increasing and
'
we show that the sequence 9x] 15
and bounded. Therefore by the definition
of monotone
convergence
{v} is converges.
thereom
this sequence
Now we need to find the limit:
*
Let
X" be the limit
lum (date) = √2+ km (x)
1im (xn+1)
7*2
+#2
-
χ
2
2+
2 + x*
x-2=0
(x*+1) (x* -2)
X=-1
= 0
x = 2
**
The limit is 2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F27ae23ce-e402-42d8-8a45-784eb26c3f7e%2Fd02ea962-e973-4dce-a847-677d9a28cf12%2Flhvixlh_processed.png&w=3840&q=75)
Transcribed Image Text:Proof by induction:
Base case:
trynna see if the sequence is increasing.
n=1
Xn €
Xn+1 by definition
X₁ = × 1+1
VEN
x₁ = X2
√ ≤ 2 ✓
Induction Steps:
Let n = k
Э
KEIN
we know
to show that
and
xx = Xxxl
we are trying
Xx+1
£
xk+2
Therefore,
+2 (πx) = (xx+1)+3
√2+x=12+ X1
Xx+1 =
Xk+2
Thus we can conclude that the sequence {X}
Sequence. by the
monotone definition
is an
increasing
Now
problem we know
we
need
to
show
if it 15
bounded
in
the
X ≤ 2
and given
√2 =
Proof by Induction:
Base case:
we know √2 ≤ X ≤ 2, UnEN
√2 ± X₁ = 2
√2 ± √2 ± 2
✓
Induction Steps: Let
n = k
B KEN
≤
and we
want to
we know √2 = Xx 2
Show that
So.
Xx+ 1 ≤ 2.
+2 (√2)+2(x)(2) +2
√2+82 €12+ x =√4
2≤ Укн ≤2
Therefore
sequence {Xx}
In conclusion
n In
base
on
this
we
know that the
is bounded.
Show that the sequence {K^} is
Therefore has the definition
we
d
bounded
In conclusion
increasing and
'
we show that the sequence 9x] 15
and bounded. Therefore by the definition
of monotone
convergence
{v} is converges.
thereom
this sequence
Now we need to find the limit:
*
Let
X" be the limit
lum (date) = √2+ km (x)
1im (xn+1)
7*2
+#2
-
χ
2
2+
2 + x*
x-2=0
(x*+1) (x* -2)
X=-1
= 0
x = 2
**
The limit is 2
Transcribed Image Text:2. Let the sequence (xn) be recursively defined by x₁ = √√2 and
Xn+1 = √2+xn, n≥1.
Show that (xn) converges and evaluate its limit.
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