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Does $b_n = a_n+1 - a_n $ is a zero sequence implies $a_n$ is convergent? [duplicate]
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Bounded sequence which is not convergent, but differences of consecutive terms converge to zero
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Let $(a_n)$ be a sequence in $mathbbR$ and let $b_n = a_n+1 - a_n $ be a zero sequence. According to my intuition, I would say that $a_n$ converges. But my solution says otherwise. How could that be? I just can't find a counterexample.
real-analysis sequences-and-series convergence
edited Jul 25 at 11:38
Shaun
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asked Jul 25 at 10:50
Anna Saabel
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marked as duplicate by Martin R, Daniel Fischer♦ real-analysis
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Jul 25 at 11:47
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
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This question already has an answer here:
Bounded sequence which is not convergent, but differences of consecutive terms converge to zero
2 answers
Let $(a_n)$ be a sequence in $mathbbR$ and let $b_n = a_n+1 - a_n $ be a zero sequence. According to my intuition, I would say that $a_n$ converges. But my solution says otherwise. How could that be? I just can't find a counterexample.
real-analysis sequences-and-series convergence
edited Jul 25 at 11:38
Shaun
7,32592972
asked Jul 25 at 10:50
Anna Saabel
1569
marked as duplicate by Martin R, Daniel Fischer♦ real-analysis
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Jul 25 at 11:47
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
1
A zero sequence? Is that one that converges to zero? Like say $sqrtn+1-sqrt n$?
– Lord Shark the Unknown
Jul 25 at 10:51
Another one: math.stackexchange.com/q/1019832/42969.
– Martin R
Jul 25 at 11:00
You have that a sequence converges if it's cauchy, since this one is a special case of cauchy $m=n+1$, one would expect that there exists such sequence so that there exists a $m$ such that $a_n-a_m$ doesn't go to $0$. Intuitively, a sequence which diverges slowly will satisfy $a_n+1-a_nto 0$.
– kingW3
Jul 25 at 11:09
Counterexample $a_n=frac1n$
– Dr. Wolfgang Hintze
Jul 25 at 13:10
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up vote
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This question already has an answer here:
Bounded sequence which is not convergent, but differences of consecutive terms converge to zero
2 answers
Let $(a_n)$ be a sequence in $mathbbR$ and let $b_n = a_n+1 - a_n $ be a zero sequence. According to my intuition, I would say that $a_n$ converges. But my solution says otherwise. How could that be? I just can't find a counterexample.
real-analysis sequences-and-series convergence
edited Jul 25 at 11:38
Shaun
7,32592972
asked Jul 25 at 10:50
Anna Saabel
1569
This question already has an answer here:
Bounded sequence which is not convergent, but differences of consecutive terms converge to zero
2 answers
Let $(a_n)$ be a sequence in $mathbbR$ and let $b_n = a_n+1 - a_n $ be a zero sequence. According to my intuition, I would say that $a_n$ converges. But my solution says otherwise. How could that be? I just can't find a counterexample.
This question already has an answer here:
Bounded sequence which is not convergent, but differences of consecutive terms converge to zero
2 answers
real-analysis sequences-and-series convergence
edited Jul 25 at 11:38
Shaun
7,32592972
asked Jul 25 at 10:50
Anna Saabel
1569
edited Jul 25 at 11:38
Shaun
7,32592972
edited Jul 25 at 11:38
Shaun
7,32592972
edited Jul 25 at 11:38
Shaun
7,32592972
7,32592972
asked Jul 25 at 10:50
Anna Saabel
1569
asked Jul 25 at 10:50
Anna Saabel
1569
asked Jul 25 at 10:50
Anna Saabel
1569
1569
marked as duplicate by Martin R, Daniel Fischer♦ real-analysis
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Jul 25 at 11:47
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
marked as duplicate by Martin R, Daniel Fischer♦ real-analysis
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Jul 25 at 11:47
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
1
A zero sequence? Is that one that converges to zero? Like say $sqrtn+1-sqrt n$?
– Lord Shark the Unknown
Jul 25 at 10:51
Another one: math.stackexchange.com/q/1019832/42969.
– Martin R
Jul 25 at 11:00
You have that a sequence converges if it's cauchy, since this one is a special case of cauchy $m=n+1$, one would expect that there exists such sequence so that there exists a $m$ such that $a_n-a_m$ doesn't go to $0$. Intuitively, a sequence which diverges slowly will satisfy $a_n+1-a_nto 0$.
– kingW3
Jul 25 at 11:09
Counterexample $a_n=frac1n$
– Dr. Wolfgang Hintze
Jul 25 at 13:10
add a comment |Â
1
A zero sequence? Is that one that converges to zero? Like say $sqrtn+1-sqrt n$?
– Lord Shark the Unknown
Jul 25 at 10:51
Another one: math.stackexchange.com/q/1019832/42969.
– Martin R
Jul 25 at 11:00
You have that a sequence converges if it's cauchy, since this one is a special case of cauchy $m=n+1$, one would expect that there exists such sequence so that there exists a $m$ such that $a_n-a_m$ doesn't go to $0$. Intuitively, a sequence which diverges slowly will satisfy $a_n+1-a_nto 0$.
– kingW3
Jul 25 at 11:09
Counterexample $a_n=frac1n$
– Dr. Wolfgang Hintze
Jul 25 at 13:10
1
1
A zero sequence? Is that one that converges to zero? Like say $sqrtn+1-sqrt n$?
– Lord Shark the Unknown
Jul 25 at 10:51
A zero sequence? Is that one that converges to zero? Like say $sqrtn+1-sqrt n$?
– Lord Shark the Unknown
Jul 25 at 10:51
Another one: math.stackexchange.com/q/1019832/42969.
– Martin R
Jul 25 at 11:00
Another one: math.stackexchange.com/q/1019832/42969.
– Martin R
Jul 25 at 11:00
You have that a sequence converges if it's cauchy, since this one is a special case of cauchy $m=n+1$, one would expect that there exists such sequence so that there exists a $m$ such that $a_n-a_m$ doesn't go to $0$. Intuitively, a sequence which diverges slowly will satisfy $a_n+1-a_nto 0$.
– kingW3
Jul 25 at 11:09
You have that a sequence converges if it's cauchy, since this one is a special case of cauchy $m=n+1$, one would expect that there exists such sequence so that there exists a $m$ such that $a_n-a_m$ doesn't go to $0$. Intuitively, a sequence which diverges slowly will satisfy $a_n+1-a_nto 0$.
– kingW3
Jul 25 at 11:09
Counterexample $a_n=frac1n$
– Dr. Wolfgang Hintze
Jul 25 at 13:10
Counterexample $a_n=frac1n$
– Dr. Wolfgang Hintze
Jul 25 at 13:10
add a comment |Â
3 Answers
3
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up vote
2
down vote
$$a_n = sum_k=1^n frac 1 k$$
answered Jul 25 at 10:52
Kenny Lau
18.5k2157
add a comment |Â
up vote
0
down vote
Counterexample: $a_n=1+frac12+...+frac1n$.
answered Jul 25 at 10:52
Fred
37.2k1237
Why the downvote ???????????????
– Fred
Jul 26 at 5:57
add a comment |Â
up vote
-1
down vote
Another nice example. $a_n = sqrtn$. Show that
$$
sqrtn+1 - sqrtn to 0qquadtextbutqquad
sqrtn to +infty
$$
answered Jul 25 at 11:23
GEdgar
58.4k264163
add a comment |Â
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
$$a_n = sum_k=1^n frac 1 k$$
answered Jul 25 at 10:52
Kenny Lau
18.5k2157
add a comment |Â
up vote
2
down vote
$$a_n = sum_k=1^n frac 1 k$$
answered Jul 25 at 10:52
Kenny Lau
18.5k2157
add a comment |Â
up vote
2
down vote
up vote
2
down vote
$$a_n = sum_k=1^n frac 1 k$$
answered Jul 25 at 10:52
Kenny Lau
18.5k2157
$$a_n = sum_k=1^n frac 1 k$$
answered Jul 25 at 10:52
Kenny Lau
18.5k2157
answered Jul 25 at 10:52
Kenny Lau
18.5k2157
answered Jul 25 at 10:52
Kenny Lau
18.5k2157
answered Jul 25 at 10:52
Kenny Lau
18.5k2157
18.5k2157
add a comment |Â
add a comment |Â
up vote
0
down vote
Counterexample: $a_n=1+frac12+...+frac1n$.
answered Jul 25 at 10:52
Fred
37.2k1237
Why the downvote ???????????????
– Fred
Jul 26 at 5:57
add a comment |Â
up vote
0
down vote
Counterexample: $a_n=1+frac12+...+frac1n$.
answered Jul 25 at 10:52
Fred
37.2k1237
Why the downvote ???????????????
– Fred
Jul 26 at 5:57
add a comment |Â
up vote
0
down vote
up vote
0
down vote
Counterexample: $a_n=1+frac12+...+frac1n$.
answered Jul 25 at 10:52
Fred
37.2k1237
Counterexample: $a_n=1+frac12+...+frac1n$.
answered Jul 25 at 10:52
Fred
37.2k1237
answered Jul 25 at 10:52
Fred
37.2k1237
answered Jul 25 at 10:52
Fred
37.2k1237
answered Jul 25 at 10:52
Fred
37.2k1237
37.2k1237
Why the downvote ???????????????
– Fred
Jul 26 at 5:57
add a comment |Â
Why the downvote ???????????????
– Fred
Jul 26 at 5:57
Why the downvote ???????????????
– Fred
Jul 26 at 5:57
Why the downvote ???????????????
– Fred
Jul 26 at 5:57
add a comment |Â
up vote
-1
down vote
Another nice example. $a_n = sqrtn$. Show that
$$
sqrtn+1 - sqrtn to 0qquadtextbutqquad
sqrtn to +infty
$$
answered Jul 25 at 11:23
GEdgar
58.4k264163
add a comment |Â
up vote
-1
down vote
Another nice example. $a_n = sqrtn$. Show that
$$
sqrtn+1 - sqrtn to 0qquadtextbutqquad
sqrtn to +infty
$$
answered Jul 25 at 11:23
GEdgar
58.4k264163
add a comment |Â
up vote
-1
down vote
up vote
-1
down vote
Another nice example. $a_n = sqrtn$. Show that
$$
sqrtn+1 - sqrtn to 0qquadtextbutqquad
sqrtn to +infty
$$
answered Jul 25 at 11:23
GEdgar
58.4k264163
Another nice example. $a_n = sqrtn$. Show that
$$
sqrtn+1 - sqrtn to 0qquadtextbutqquad
sqrtn to +infty
$$
answered Jul 25 at 11:23
GEdgar
58.4k264163
answered Jul 25 at 11:23
GEdgar
58.4k264163
answered Jul 25 at 11:23
GEdgar
58.4k264163
answered Jul 25 at 11:23
GEdgar
58.4k264163
58.4k264163
add a comment |Â
add a comment |Â
1
A zero sequence? Is that one that converges to zero? Like say $sqrtn+1-sqrt n$?
– Lord Shark the Unknown
Jul 25 at 10:51
Another one: math.stackexchange.com/q/1019832/42969.
– Martin R
Jul 25 at 11:00
You have that a sequence converges if it's cauchy, since this one is a special case of cauchy $m=n+1$, one would expect that there exists such sequence so that there exists a $m$ such that $a_n-a_m$ doesn't go to $0$. Intuitively, a sequence which diverges slowly will satisfy $a_n+1-a_nto 0$.
– kingW3
Jul 25 at 11:09
Counterexample $a_n=frac1n$
– Dr. Wolfgang Hintze
Jul 25 at 13:10