Proof for Convergence of Nested Sequences
With this being my first week learning about limits, their rules, and convergence/divergence, I received another question:
$a_k$ is a convergent sequence with $lim_{nto infty}(a_k) = a$
Let $p(x) = 3x^2 - x + 2$. Show that $(p(a_k))$ is also convergent with $lim_{kto infty}p(a_k) = p(a)$
My approach would be to use the multiplication rule, which is: when you have two convergent sequences $lim_{nto infty}(a_n) = a$ and $lim_{nto infty}(b_n) = b$, then
$lim_{nto infty}(a_n cdot b_n) = lim_{nto infty}(a_n) cdot lim_{nto infty}(b_n) = a cdot b$
But I'm not entirely sure about how to apply that rule specifically to this question. I get confused when so many variables are used. I also thought about manually listing out both sequences and looking at $a_k$ as a subsequence to $p(x)$, yet I'm not certain if that'd work.
I hope someone can clarify the best approach to both parts and explain their answer in detail.
Thank you!!
sequences-and-series limits convergence proof-explanation
asked Dec 6 at 17:28
Rikk
424
add a comment |
With this being my first week learning about limits, their rules, and convergence/divergence, I received another question:
$a_k$ is a convergent sequence with $lim_{nto infty}(a_k) = a$
Let $p(x) = 3x^2 - x + 2$. Show that $(p(a_k))$ is also convergent with $lim_{kto infty}p(a_k) = p(a)$
My approach would be to use the multiplication rule, which is: when you have two convergent sequences $lim_{nto infty}(a_n) = a$ and $lim_{nto infty}(b_n) = b$, then
$lim_{nto infty}(a_n cdot b_n) = lim_{nto infty}(a_n) cdot lim_{nto infty}(b_n) = a cdot b$
But I'm not entirely sure about how to apply that rule specifically to this question. I get confused when so many variables are used. I also thought about manually listing out both sequences and looking at $a_k$ as a subsequence to $p(x)$, yet I'm not certain if that'd work.
I hope someone can clarify the best approach to both parts and explain their answer in detail.
Thank you!!
sequences-and-series limits convergence proof-explanation
asked Dec 6 at 17:28
Rikk
424
add a comment |
With this being my first week learning about limits, their rules, and convergence/divergence, I received another question:
$a_k$ is a convergent sequence with $lim_{nto infty}(a_k) = a$
Let $p(x) = 3x^2 - x + 2$. Show that $(p(a_k))$ is also convergent with $lim_{kto infty}p(a_k) = p(a)$
My approach would be to use the multiplication rule, which is: when you have two convergent sequences $lim_{nto infty}(a_n) = a$ and $lim_{nto infty}(b_n) = b$, then
$lim_{nto infty}(a_n cdot b_n) = lim_{nto infty}(a_n) cdot lim_{nto infty}(b_n) = a cdot b$
But I'm not entirely sure about how to apply that rule specifically to this question. I get confused when so many variables are used. I also thought about manually listing out both sequences and looking at $a_k$ as a subsequence to $p(x)$, yet I'm not certain if that'd work.
I hope someone can clarify the best approach to both parts and explain their answer in detail.
Thank you!!
sequences-and-series limits convergence proof-explanation
asked Dec 6 at 17:28
Rikk
424
With this being my first week learning about limits, their rules, and convergence/divergence, I received another question:
$a_k$ is a convergent sequence with $lim_{nto infty}(a_k) = a$
Let $p(x) = 3x^2 - x + 2$. Show that $(p(a_k))$ is also convergent with $lim_{kto infty}p(a_k) = p(a)$
My approach would be to use the multiplication rule, which is: when you have two convergent sequences $lim_{nto infty}(a_n) = a$ and $lim_{nto infty}(b_n) = b$, then
$lim_{nto infty}(a_n cdot b_n) = lim_{nto infty}(a_n) cdot lim_{nto infty}(b_n) = a cdot b$
But I'm not entirely sure about how to apply that rule specifically to this question. I get confused when so many variables are used. I also thought about manually listing out both sequences and looking at $a_k$ as a subsequence to $p(x)$, yet I'm not certain if that'd work.
I hope someone can clarify the best approach to both parts and explain their answer in detail.
Thank you!!
sequences-and-series limits convergence proof-explanation
sequences-and-series limits convergence proof-explanation
asked Dec 6 at 17:28
Rikk
424
asked Dec 6 at 17:28
Rikk
424
edited Dec 6 at 22:04
asked Dec 6 at 17:28
Rikk
424
asked Dec 6 at 17:28
Rikk
424
asked Dec 6 at 17:28
Rikk
424
424
add a comment |
add a comment |
2 Answers
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Have you gone over the definition of continuity for real valued functions in your course yet?
If so you can use the continuity of the polynomials in (a) and (b) to prove the result.
Otherwise your approach for using sums and multiplication of sequences will work. Indeed you can rewrite your sequence $p(a_k)$ as $p(a_k) = 3 times a_k times a_k -a_k +2$.
From here can you see where adding and multiplying sequences will give you what you want?
answered Dec 6 at 17:43
C. Grant
1114
add a comment |
You are on the right track. In part a)
$$p(a_k)= 3a_k^2-a_k+2$$
Now you need to use various rules, like the product rule and the sum rule. For the first term, $$lim_{ktoinfty}a_k^2=lim_{ktoinfty}a_kcdotlim_{ktoinfty}a_k=a^2$$
Once you complete part a) you should find part b) simple.
answered Dec 6 at 17:43
saulspatz
13.8k21328
Could you still go over each part in detail so I can follow? I'm still not seeing it as clear as I'd like to :/ Thanks!
– Rikk
Dec 6 at 21:08
@Rikk It would help if you edit your question to show exactly what you understand, and where you are stuck. Otherwise, I don't know what else to say.
– saulspatz
Dec 6 at 21:11
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
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active
oldest
votes
Have you gone over the definition of continuity for real valued functions in your course yet?
If so you can use the continuity of the polynomials in (a) and (b) to prove the result.
Otherwise your approach for using sums and multiplication of sequences will work. Indeed you can rewrite your sequence $p(a_k)$ as $p(a_k) = 3 times a_k times a_k -a_k +2$.
From here can you see where adding and multiplying sequences will give you what you want?
answered Dec 6 at 17:43
C. Grant
1114
add a comment |
Have you gone over the definition of continuity for real valued functions in your course yet?
If so you can use the continuity of the polynomials in (a) and (b) to prove the result.
Otherwise your approach for using sums and multiplication of sequences will work. Indeed you can rewrite your sequence $p(a_k)$ as $p(a_k) = 3 times a_k times a_k -a_k +2$.
From here can you see where adding and multiplying sequences will give you what you want?
answered Dec 6 at 17:43
C. Grant
1114
add a comment |
Have you gone over the definition of continuity for real valued functions in your course yet?
If so you can use the continuity of the polynomials in (a) and (b) to prove the result.
Otherwise your approach for using sums and multiplication of sequences will work. Indeed you can rewrite your sequence $p(a_k)$ as $p(a_k) = 3 times a_k times a_k -a_k +2$.
From here can you see where adding and multiplying sequences will give you what you want?
answered Dec 6 at 17:43
C. Grant
1114
Have you gone over the definition of continuity for real valued functions in your course yet?
If so you can use the continuity of the polynomials in (a) and (b) to prove the result.
Otherwise your approach for using sums and multiplication of sequences will work. Indeed you can rewrite your sequence $p(a_k)$ as $p(a_k) = 3 times a_k times a_k -a_k +2$.
From here can you see where adding and multiplying sequences will give you what you want?
answered Dec 6 at 17:43
C. Grant
1114
answered Dec 6 at 17:43
C. Grant
1114
answered Dec 6 at 17:43
C. Grant
1114
answered Dec 6 at 17:43
C. Grant
1114
1114
add a comment |
add a comment |
You are on the right track. In part a)
$$p(a_k)= 3a_k^2-a_k+2$$
Now you need to use various rules, like the product rule and the sum rule. For the first term, $$lim_{ktoinfty}a_k^2=lim_{ktoinfty}a_kcdotlim_{ktoinfty}a_k=a^2$$
Once you complete part a) you should find part b) simple.
answered Dec 6 at 17:43
saulspatz
13.8k21328
Could you still go over each part in detail so I can follow? I'm still not seeing it as clear as I'd like to :/ Thanks!
– Rikk
Dec 6 at 21:08
@Rikk It would help if you edit your question to show exactly what you understand, and where you are stuck. Otherwise, I don't know what else to say.
– saulspatz
Dec 6 at 21:11
add a comment |
You are on the right track. In part a)
$$p(a_k)= 3a_k^2-a_k+2$$
Now you need to use various rules, like the product rule and the sum rule. For the first term, $$lim_{ktoinfty}a_k^2=lim_{ktoinfty}a_kcdotlim_{ktoinfty}a_k=a^2$$
Once you complete part a) you should find part b) simple.
answered Dec 6 at 17:43
saulspatz
13.8k21328
Could you still go over each part in detail so I can follow? I'm still not seeing it as clear as I'd like to :/ Thanks!
– Rikk
Dec 6 at 21:08
@Rikk It would help if you edit your question to show exactly what you understand, and where you are stuck. Otherwise, I don't know what else to say.
– saulspatz
Dec 6 at 21:11
add a comment |
You are on the right track. In part a)
$$p(a_k)= 3a_k^2-a_k+2$$
Now you need to use various rules, like the product rule and the sum rule. For the first term, $$lim_{ktoinfty}a_k^2=lim_{ktoinfty}a_kcdotlim_{ktoinfty}a_k=a^2$$
Once you complete part a) you should find part b) simple.
answered Dec 6 at 17:43
saulspatz
13.8k21328
You are on the right track. In part a)
$$p(a_k)= 3a_k^2-a_k+2$$
Now you need to use various rules, like the product rule and the sum rule. For the first term, $$lim_{ktoinfty}a_k^2=lim_{ktoinfty}a_kcdotlim_{ktoinfty}a_k=a^2$$
Once you complete part a) you should find part b) simple.
answered Dec 6 at 17:43
saulspatz
13.8k21328
answered Dec 6 at 17:43
saulspatz
13.8k21328
answered Dec 6 at 17:43
saulspatz
13.8k21328
answered Dec 6 at 17:43
saulspatz
13.8k21328
13.8k21328
Could you still go over each part in detail so I can follow? I'm still not seeing it as clear as I'd like to :/ Thanks!
– Rikk
Dec 6 at 21:08
@Rikk It would help if you edit your question to show exactly what you understand, and where you are stuck. Otherwise, I don't know what else to say.
– saulspatz
Dec 6 at 21:11
add a comment |
Could you still go over each part in detail so I can follow? I'm still not seeing it as clear as I'd like to :/ Thanks!
– Rikk
Dec 6 at 21:08
@Rikk It would help if you edit your question to show exactly what you understand, and where you are stuck. Otherwise, I don't know what else to say.
– saulspatz
Dec 6 at 21:11
Could you still go over each part in detail so I can follow? I'm still not seeing it as clear as I'd like to :/ Thanks!
– Rikk
Dec 6 at 21:08
Could you still go over each part in detail so I can follow? I'm still not seeing it as clear as I'd like to :/ Thanks!
– Rikk
Dec 6 at 21:08
@Rikk It would help if you edit your question to show exactly what you understand, and where you are stuck. Otherwise, I don't know what else to say.
– saulspatz
Dec 6 at 21:11
@Rikk It would help if you edit your question to show exactly what you understand, and where you are stuck. Otherwise, I don't know what else to say.
– saulspatz
Dec 6 at 21:11
add a comment |
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