Convince me: limit of sum of a constant is infinity
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So I have a problem and have simplified the part I am confused about below.
If $sum_{m=1}^{infty }c < infty$ and $0 leq c leq 1$, then $lim_{nrightarrow infty} sum_{m=n}^{infty }c= 0$ which implies $c=0$.
My general intuition says that because the sum of infinitely many non-negative c's is less than infinity, than $c=0$ because the sum of an infinitely many positive numbers will always be infinity.
The limit is where I am confused. I feel like the limit will always be $0$ even if $c>0$. It also feels like the limit is not necessary to show $c=0$.
limits summation infinity
asked Nov 15 at 5:03
kpr62
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So I have a problem and have simplified the part I am confused about below.
If $sum_{m=1}^{infty }c < infty$ and $0 leq c leq 1$, then $lim_{nrightarrow infty} sum_{m=n}^{infty }c= 0$ which implies $c=0$.
My general intuition says that because the sum of infinitely many non-negative c's is less than infinity, than $c=0$ because the sum of an infinitely many positive numbers will always be infinity.
The limit is where I am confused. I feel like the limit will always be $0$ even if $c>0$. It also feels like the limit is not necessary to show $c=0$.
limits summation infinity
asked Nov 15 at 5:03
kpr62
103
New contributor
kpr62 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
I think you are missing something in your question. Are you really dealing with a constant. Or are you dealing (for example when dealing with integration) a function of some sort?
– Q the Platypus
Nov 15 at 5:13
I think you are confused with "fixing" the value of $c$ first since you are talking about constants. If $c = 0$ has been fixed beforehand, of course the sum is zero but if $c>0$ is chosen instead, then you will tend to infinity by taking the limit of partial sum of the series. You can see clearly the limit tends to infinity simply by looking at the definition of limit.
– Evan William Chandra
Nov 15 at 5:16
I'm not actually dealing with a constant. I have a sequence {X_n} in which a term in the sequence occurs infinitely often, say {X_i}. I then have a function for each term f(X_j) in which the function 0<=f(X_j)<=1. for all X_j. The question is asking us to show that f(X_i), which occurs infinitely often, must be equal to zero if the sum of all f(X_j) is less than infinity.
– kpr62
Nov 15 at 5:22
The solution uses a limit, and I don't understand why.
– kpr62
Nov 15 at 5:24
@kpr The sum of your subsequence of infinitely occurring $f(X_i)$s is a lower bound for the sum of the whole sequence. You're right that the problem reduces to what you stated in your question.
– Alexander Gruber♦
Nov 15 at 5:25
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
So I have a problem and have simplified the part I am confused about below.
If $sum_{m=1}^{infty }c < infty$ and $0 leq c leq 1$, then $lim_{nrightarrow infty} sum_{m=n}^{infty }c= 0$ which implies $c=0$.
My general intuition says that because the sum of infinitely many non-negative c's is less than infinity, than $c=0$ because the sum of an infinitely many positive numbers will always be infinity.
The limit is where I am confused. I feel like the limit will always be $0$ even if $c>0$. It also feels like the limit is not necessary to show $c=0$.
limits summation infinity
asked Nov 15 at 5:03
kpr62
103
New contributor
kpr62 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
So I have a problem and have simplified the part I am confused about below.
If $sum_{m=1}^{infty }c < infty$ and $0 leq c leq 1$, then $lim_{nrightarrow infty} sum_{m=n}^{infty }c= 0$ which implies $c=0$.
My general intuition says that because the sum of infinitely many non-negative c's is less than infinity, than $c=0$ because the sum of an infinitely many positive numbers will always be infinity.
The limit is where I am confused. I feel like the limit will always be $0$ even if $c>0$. It also feels like the limit is not necessary to show $c=0$.
limits summation infinity
limits summation infinity
asked Nov 15 at 5:03
kpr62
103
New contributor
kpr62 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Nov 15 at 5:03
kpr62
103
New contributor
kpr62 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Nov 15 at 5:03
kpr62
103
New contributor
kpr62 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Nov 15 at 5:03
kpr62
103
asked Nov 15 at 5:03
kpr62
103
103
New contributor
kpr62 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
kpr62 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
kpr62 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
I think you are missing something in your question. Are you really dealing with a constant. Or are you dealing (for example when dealing with integration) a function of some sort?
– Q the Platypus
Nov 15 at 5:13
I think you are confused with "fixing" the value of $c$ first since you are talking about constants. If $c = 0$ has been fixed beforehand, of course the sum is zero but if $c>0$ is chosen instead, then you will tend to infinity by taking the limit of partial sum of the series. You can see clearly the limit tends to infinity simply by looking at the definition of limit.
– Evan William Chandra
Nov 15 at 5:16
I'm not actually dealing with a constant. I have a sequence {X_n} in which a term in the sequence occurs infinitely often, say {X_i}. I then have a function for each term f(X_j) in which the function 0<=f(X_j)<=1. for all X_j. The question is asking us to show that f(X_i), which occurs infinitely often, must be equal to zero if the sum of all f(X_j) is less than infinity.
– kpr62
Nov 15 at 5:22
The solution uses a limit, and I don't understand why.
– kpr62
Nov 15 at 5:24
@kpr The sum of your subsequence of infinitely occurring $f(X_i)$s is a lower bound for the sum of the whole sequence. You're right that the problem reduces to what you stated in your question.
– Alexander Gruber♦
Nov 15 at 5:25
add a comment |
I think you are missing something in your question. Are you really dealing with a constant. Or are you dealing (for example when dealing with integration) a function of some sort?
– Q the Platypus
Nov 15 at 5:13
I think you are confused with "fixing" the value of $c$ first since you are talking about constants. If $c = 0$ has been fixed beforehand, of course the sum is zero but if $c>0$ is chosen instead, then you will tend to infinity by taking the limit of partial sum of the series. You can see clearly the limit tends to infinity simply by looking at the definition of limit.
– Evan William Chandra
Nov 15 at 5:16
I'm not actually dealing with a constant. I have a sequence {X_n} in which a term in the sequence occurs infinitely often, say {X_i}. I then have a function for each term f(X_j) in which the function 0<=f(X_j)<=1. for all X_j. The question is asking us to show that f(X_i), which occurs infinitely often, must be equal to zero if the sum of all f(X_j) is less than infinity.
– kpr62
Nov 15 at 5:22
The solution uses a limit, and I don't understand why.
– kpr62
Nov 15 at 5:24
@kpr The sum of your subsequence of infinitely occurring $f(X_i)$s is a lower bound for the sum of the whole sequence. You're right that the problem reduces to what you stated in your question.
– Alexander Gruber♦
Nov 15 at 5:25
I think you are missing something in your question. Are you really dealing with a constant. Or are you dealing (for example when dealing with integration) a function of some sort?
– Q the Platypus
Nov 15 at 5:13
I think you are missing something in your question. Are you really dealing with a constant. Or are you dealing (for example when dealing with integration) a function of some sort?
– Q the Platypus
Nov 15 at 5:13
I think you are confused with "fixing" the value of $c$ first since you are talking about constants. If $c = 0$ has been fixed beforehand, of course the sum is zero but if $c>0$ is chosen instead, then you will tend to infinity by taking the limit of partial sum of the series. You can see clearly the limit tends to infinity simply by looking at the definition of limit.
– Evan William Chandra
Nov 15 at 5:16
I think you are confused with "fixing" the value of $c$ first since you are talking about constants. If $c = 0$ has been fixed beforehand, of course the sum is zero but if $c>0$ is chosen instead, then you will tend to infinity by taking the limit of partial sum of the series. You can see clearly the limit tends to infinity simply by looking at the definition of limit.
– Evan William Chandra
Nov 15 at 5:16
I'm not actually dealing with a constant. I have a sequence {X_n} in which a term in the sequence occurs infinitely often, say {X_i}. I then have a function for each term f(X_j) in which the function 0<=f(X_j)<=1. for all X_j. The question is asking us to show that f(X_i), which occurs infinitely often, must be equal to zero if the sum of all f(X_j) is less than infinity.
– kpr62
Nov 15 at 5:22
I'm not actually dealing with a constant. I have a sequence {X_n} in which a term in the sequence occurs infinitely often, say {X_i}. I then have a function for each term f(X_j) in which the function 0<=f(X_j)<=1. for all X_j. The question is asking us to show that f(X_i), which occurs infinitely often, must be equal to zero if the sum of all f(X_j) is less than infinity.
– kpr62
Nov 15 at 5:22
The solution uses a limit, and I don't understand why.
– kpr62
Nov 15 at 5:24
The solution uses a limit, and I don't understand why.
– kpr62
Nov 15 at 5:24
@kpr The sum of your subsequence of infinitely occurring $f(X_i)$s is a lower bound for the sum of the whole sequence. You're right that the problem reduces to what you stated in your question.
– Alexander Gruber♦
Nov 15 at 5:25
@kpr The sum of your subsequence of infinitely occurring $f(X_i)$s is a lower bound for the sum of the whole sequence. You're right that the problem reduces to what you stated in your question.
– Alexander Gruber♦
Nov 15 at 5:25
add a comment |
3 Answers
3
active
oldest
votes
up vote
2
down vote
accepted
If $c>0$ then $sum_{i=1}^{infty }c= lim_{nto infty }
sum_{i=1}^{n}c=lim_{nto infty }nc =infty $
If c=$0$ then $sum_{i=1}^{infty }c=0 $
If $c<0 $then $sum_{i=1}^{infty }c =-infty $
answered Nov 15 at 5:27
Dadrahm
3288
Add this with @Alexander Gruber's comment, and I am convinced.
– kpr62
Nov 15 at 5:28
Thanks for the help!
– kpr62
Nov 15 at 5:29
add a comment |
up vote
1
down vote
Suppose that every day I go to the bank and deposit the same amount of money: $c$ dollars.
I want to buy a gold chain that costs $M$ dollars. Eventually, if I am diligent and keep depositing $c$ dollars every day, I will have enough to buy my gold chain, right? No matter how much $M$ is.
The only way this doesn't work is if the amount of dollars I am depositing every day is $0$.
answered Nov 15 at 5:16
Alexander Gruber♦
20.1k24102171
it is a good example/explination. But I think you should explain how your example connects to the definition of having a limit equal infinity.
– Q the Platypus
Nov 15 at 5:18
add a comment |
up vote
0
down vote
Since your sum $$sum _1^infty C <infty$$
We may apply the divergence test to conclude that $$lim _ {nto infty}C=0.$$
Since C is a constant we have $$lim _{nto infty }C =C.$$
Thus $C=0.$
answered Nov 15 at 5:29
Mohammad Riazi-Kermani
39.8k41957
add a comment |
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
If $c>0$ then $sum_{i=1}^{infty }c= lim_{nto infty }
sum_{i=1}^{n}c=lim_{nto infty }nc =infty $
If c=$0$ then $sum_{i=1}^{infty }c=0 $
If $c<0 $then $sum_{i=1}^{infty }c =-infty $
answered Nov 15 at 5:27
Dadrahm
3288
Add this with @Alexander Gruber's comment, and I am convinced.
– kpr62
Nov 15 at 5:28
Thanks for the help!
– kpr62
Nov 15 at 5:29
add a comment |
up vote
2
down vote
accepted
If $c>0$ then $sum_{i=1}^{infty }c= lim_{nto infty }
sum_{i=1}^{n}c=lim_{nto infty }nc =infty $
If c=$0$ then $sum_{i=1}^{infty }c=0 $
If $c<0 $then $sum_{i=1}^{infty }c =-infty $
answered Nov 15 at 5:27
Dadrahm
3288
Add this with @Alexander Gruber's comment, and I am convinced.
– kpr62
Nov 15 at 5:28
Thanks for the help!
– kpr62
Nov 15 at 5:29
add a comment |
up vote
2
down vote
accepted
up vote
2
down vote
accepted
If $c>0$ then $sum_{i=1}^{infty }c= lim_{nto infty }
sum_{i=1}^{n}c=lim_{nto infty }nc =infty $
If c=$0$ then $sum_{i=1}^{infty }c=0 $
If $c<0 $then $sum_{i=1}^{infty }c =-infty $
answered Nov 15 at 5:27
Dadrahm
3288
If $c>0$ then $sum_{i=1}^{infty }c= lim_{nto infty }
sum_{i=1}^{n}c=lim_{nto infty }nc =infty $
If c=$0$ then $sum_{i=1}^{infty }c=0 $
If $c<0 $then $sum_{i=1}^{infty }c =-infty $
answered Nov 15 at 5:27
Dadrahm
3288
answered Nov 15 at 5:27
Dadrahm
3288
answered Nov 15 at 5:27
Dadrahm
3288
answered Nov 15 at 5:27
Dadrahm
3288
3288
Add this with @Alexander Gruber's comment, and I am convinced.
– kpr62
Nov 15 at 5:28
Thanks for the help!
– kpr62
Nov 15 at 5:29
add a comment |
Add this with @Alexander Gruber's comment, and I am convinced.
– kpr62
Nov 15 at 5:28
Thanks for the help!
– kpr62
Nov 15 at 5:29
Add this with @Alexander Gruber's comment, and I am convinced.
– kpr62
Nov 15 at 5:28
Add this with @Alexander Gruber's comment, and I am convinced.
– kpr62
Nov 15 at 5:28
Thanks for the help!
– kpr62
Nov 15 at 5:29
Thanks for the help!
– kpr62
Nov 15 at 5:29
add a comment |
up vote
1
down vote
Suppose that every day I go to the bank and deposit the same amount of money: $c$ dollars.
I want to buy a gold chain that costs $M$ dollars. Eventually, if I am diligent and keep depositing $c$ dollars every day, I will have enough to buy my gold chain, right? No matter how much $M$ is.
The only way this doesn't work is if the amount of dollars I am depositing every day is $0$.
answered Nov 15 at 5:16
Alexander Gruber♦
20.1k24102171
it is a good example/explination. But I think you should explain how your example connects to the definition of having a limit equal infinity.
– Q the Platypus
Nov 15 at 5:18
add a comment |
up vote
1
down vote
Suppose that every day I go to the bank and deposit the same amount of money: $c$ dollars.
I want to buy a gold chain that costs $M$ dollars. Eventually, if I am diligent and keep depositing $c$ dollars every day, I will have enough to buy my gold chain, right? No matter how much $M$ is.
The only way this doesn't work is if the amount of dollars I am depositing every day is $0$.
answered Nov 15 at 5:16
Alexander Gruber♦
20.1k24102171
it is a good example/explination. But I think you should explain how your example connects to the definition of having a limit equal infinity.
– Q the Platypus
Nov 15 at 5:18
add a comment |
up vote
1
down vote
up vote
1
down vote
Suppose that every day I go to the bank and deposit the same amount of money: $c$ dollars.
I want to buy a gold chain that costs $M$ dollars. Eventually, if I am diligent and keep depositing $c$ dollars every day, I will have enough to buy my gold chain, right? No matter how much $M$ is.
The only way this doesn't work is if the amount of dollars I am depositing every day is $0$.
answered Nov 15 at 5:16
Alexander Gruber♦
20.1k24102171
Suppose that every day I go to the bank and deposit the same amount of money: $c$ dollars.
I want to buy a gold chain that costs $M$ dollars. Eventually, if I am diligent and keep depositing $c$ dollars every day, I will have enough to buy my gold chain, right? No matter how much $M$ is.
The only way this doesn't work is if the amount of dollars I am depositing every day is $0$.
answered Nov 15 at 5:16
Alexander Gruber♦
20.1k24102171
answered Nov 15 at 5:16
Alexander Gruber♦
20.1k24102171
answered Nov 15 at 5:16
Alexander Gruber♦
20.1k24102171
answered Nov 15 at 5:16
Alexander Gruber♦
20.1k24102171
20.1k24102171
it is a good example/explination. But I think you should explain how your example connects to the definition of having a limit equal infinity.
– Q the Platypus
Nov 15 at 5:18
add a comment |
it is a good example/explination. But I think you should explain how your example connects to the definition of having a limit equal infinity.
– Q the Platypus
Nov 15 at 5:18
it is a good example/explination. But I think you should explain how your example connects to the definition of having a limit equal infinity.
– Q the Platypus
Nov 15 at 5:18
it is a good example/explination. But I think you should explain how your example connects to the definition of having a limit equal infinity.
– Q the Platypus
Nov 15 at 5:18
add a comment |
up vote
0
down vote
Since your sum $$sum _1^infty C <infty$$
We may apply the divergence test to conclude that $$lim _ {nto infty}C=0.$$
Since C is a constant we have $$lim _{nto infty }C =C.$$
Thus $C=0.$
answered Nov 15 at 5:29
Mohammad Riazi-Kermani
39.8k41957
add a comment |
up vote
0
down vote
Since your sum $$sum _1^infty C <infty$$
We may apply the divergence test to conclude that $$lim _ {nto infty}C=0.$$
Since C is a constant we have $$lim _{nto infty }C =C.$$
Thus $C=0.$
answered Nov 15 at 5:29
Mohammad Riazi-Kermani
39.8k41957
add a comment |
up vote
0
down vote
up vote
0
down vote
Since your sum $$sum _1^infty C <infty$$
We may apply the divergence test to conclude that $$lim _ {nto infty}C=0.$$
Since C is a constant we have $$lim _{nto infty }C =C.$$
Thus $C=0.$
answered Nov 15 at 5:29
Mohammad Riazi-Kermani
39.8k41957
Since your sum $$sum _1^infty C <infty$$
We may apply the divergence test to conclude that $$lim _ {nto infty}C=0.$$
Since C is a constant we have $$lim _{nto infty }C =C.$$
Thus $C=0.$
answered Nov 15 at 5:29
Mohammad Riazi-Kermani
39.8k41957
answered Nov 15 at 5:29
Mohammad Riazi-Kermani
39.8k41957
answered Nov 15 at 5:29
Mohammad Riazi-Kermani
39.8k41957
answered Nov 15 at 5:29
Mohammad Riazi-Kermani
39.8k41957
39.8k41957
add a comment |
add a comment |
kpr62 is a new contributor. Be nice, and check out our Code of Conduct.
kpr62 is a new contributor. Be nice, and check out our Code of Conduct.
kpr62 is a new contributor. Be nice, and check out our Code of Conduct.
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I think you are missing something in your question. Are you really dealing with a constant. Or are you dealing (for example when dealing with integration) a function of some sort?
– Q the Platypus
Nov 15 at 5:13
I think you are confused with "fixing" the value of $c$ first since you are talking about constants. If $c = 0$ has been fixed beforehand, of course the sum is zero but if $c>0$ is chosen instead, then you will tend to infinity by taking the limit of partial sum of the series. You can see clearly the limit tends to infinity simply by looking at the definition of limit.
– Evan William Chandra
Nov 15 at 5:16
I'm not actually dealing with a constant. I have a sequence {X_n} in which a term in the sequence occurs infinitely often, say {X_i}. I then have a function for each term f(X_j) in which the function 0<=f(X_j)<=1. for all X_j. The question is asking us to show that f(X_i), which occurs infinitely often, must be equal to zero if the sum of all f(X_j) is less than infinity.
– kpr62
Nov 15 at 5:22
The solution uses a limit, and I don't understand why.
– kpr62
Nov 15 at 5:24
@kpr The sum of your subsequence of infinitely occurring $f(X_i)$s is a lower bound for the sum of the whole sequence. You're right that the problem reduces to what you stated in your question.
– Alexander Gruber♦
Nov 15 at 5:25