Diagonal subsequence converges at all points under these conditions
I do not understand why the diagonal subsequence $g_n = f_{n,n}$ converges at any point $w_j$. for example, because $g_{n} (w_{20})$ it converges? From what I understand we would have ${ f_{1,1} (w_{20}), f_{2,2} (w_{20}) , f_{3,3} (w_{20}), ... }$. Why does it converge? If I'm wrong, someone can explain?
complex-analysis
asked Nov 30 at 0:29
Ricardo Freire
392110
add a comment |
I do not understand why the diagonal subsequence $g_n = f_{n,n}$ converges at any point $w_j$. for example, because $g_{n} (w_{20})$ it converges? From what I understand we would have ${ f_{1,1} (w_{20}), f_{2,2} (w_{20}) , f_{3,3} (w_{20}), ... }$. Why does it converge? If I'm wrong, someone can explain?
complex-analysis
asked Nov 30 at 0:29
Ricardo Freire
392110
add a comment |
I do not understand why the diagonal subsequence $g_n = f_{n,n}$ converges at any point $w_j$. for example, because $g_{n} (w_{20})$ it converges? From what I understand we would have ${ f_{1,1} (w_{20}), f_{2,2} (w_{20}) , f_{3,3} (w_{20}), ... }$. Why does it converge? If I'm wrong, someone can explain?
complex-analysis
asked Nov 30 at 0:29
Ricardo Freire
392110
I do not understand why the diagonal subsequence $g_n = f_{n,n}$ converges at any point $w_j$. for example, because $g_{n} (w_{20})$ it converges? From what I understand we would have ${ f_{1,1} (w_{20}), f_{2,2} (w_{20}) , f_{3,3} (w_{20}), ... }$. Why does it converge? If I'm wrong, someone can explain?
complex-analysis
complex-analysis
asked Nov 30 at 0:29
Ricardo Freire
392110
asked Nov 30 at 0:29
Ricardo Freire
392110
asked Nov 30 at 0:29
Ricardo Freire
392110
asked Nov 30 at 0:29
Ricardo Freire
392110
asked Nov 30 at 0:29
Ricardo Freire
392110
392110
add a comment |
add a comment |
1 Answer
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Except for the first 19 terms, the sequence $(g_n)$ is a subsequence of $(f_{n,20})$, which converges at $w_{20}$.
answered Nov 30 at 0:33
Andreas Blass
49.1k351106
More suppose the case $g_3(w_5)$, we have by the construction that $ g_5(w_5) $ is convergent, and $ g_5 (w_5) $ is only a subsequence of $ g_3 (w_5)$. How to guarantee the convergence of $ g_3 (w_5)$ entire?
– Ricardo Freire
Nov 30 at 3:48
@RicardoFreire I don't understand your comment, since $g_3(w_5)$ isn't a sequence.
– Andreas Blass
Nov 30 at 12:19
add a comment |
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1 Answer
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1 Answer
1
active
oldest
votes
active
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active
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votes
Except for the first 19 terms, the sequence $(g_n)$ is a subsequence of $(f_{n,20})$, which converges at $w_{20}$.
answered Nov 30 at 0:33
Andreas Blass
49.1k351106
More suppose the case $g_3(w_5)$, we have by the construction that $ g_5(w_5) $ is convergent, and $ g_5 (w_5) $ is only a subsequence of $ g_3 (w_5)$. How to guarantee the convergence of $ g_3 (w_5)$ entire?
– Ricardo Freire
Nov 30 at 3:48
@RicardoFreire I don't understand your comment, since $g_3(w_5)$ isn't a sequence.
– Andreas Blass
Nov 30 at 12:19
add a comment |
Except for the first 19 terms, the sequence $(g_n)$ is a subsequence of $(f_{n,20})$, which converges at $w_{20}$.
answered Nov 30 at 0:33
Andreas Blass
49.1k351106
More suppose the case $g_3(w_5)$, we have by the construction that $ g_5(w_5) $ is convergent, and $ g_5 (w_5) $ is only a subsequence of $ g_3 (w_5)$. How to guarantee the convergence of $ g_3 (w_5)$ entire?
– Ricardo Freire
Nov 30 at 3:48
@RicardoFreire I don't understand your comment, since $g_3(w_5)$ isn't a sequence.
– Andreas Blass
Nov 30 at 12:19
add a comment |
Except for the first 19 terms, the sequence $(g_n)$ is a subsequence of $(f_{n,20})$, which converges at $w_{20}$.
answered Nov 30 at 0:33
Andreas Blass
49.1k351106
Except for the first 19 terms, the sequence $(g_n)$ is a subsequence of $(f_{n,20})$, which converges at $w_{20}$.
answered Nov 30 at 0:33
Andreas Blass
49.1k351106
answered Nov 30 at 0:33
Andreas Blass
49.1k351106
answered Nov 30 at 0:33
Andreas Blass
49.1k351106
answered Nov 30 at 0:33
Andreas Blass
49.1k351106
49.1k351106
More suppose the case $g_3(w_5)$, we have by the construction that $ g_5(w_5) $ is convergent, and $ g_5 (w_5) $ is only a subsequence of $ g_3 (w_5)$. How to guarantee the convergence of $ g_3 (w_5)$ entire?
– Ricardo Freire
Nov 30 at 3:48
@RicardoFreire I don't understand your comment, since $g_3(w_5)$ isn't a sequence.
– Andreas Blass
Nov 30 at 12:19
add a comment |
More suppose the case $g_3(w_5)$, we have by the construction that $ g_5(w_5) $ is convergent, and $ g_5 (w_5) $ is only a subsequence of $ g_3 (w_5)$. How to guarantee the convergence of $ g_3 (w_5)$ entire?
– Ricardo Freire
Nov 30 at 3:48
@RicardoFreire I don't understand your comment, since $g_3(w_5)$ isn't a sequence.
– Andreas Blass
Nov 30 at 12:19
More suppose the case $g_3(w_5)$, we have by the construction that $ g_5(w_5) $ is convergent, and $ g_5 (w_5) $ is only a subsequence of $ g_3 (w_5)$. How to guarantee the convergence of $ g_3 (w_5)$ entire?
– Ricardo Freire
Nov 30 at 3:48
More suppose the case $g_3(w_5)$, we have by the construction that $ g_5(w_5) $ is convergent, and $ g_5 (w_5) $ is only a subsequence of $ g_3 (w_5)$. How to guarantee the convergence of $ g_3 (w_5)$ entire?
– Ricardo Freire
Nov 30 at 3:48
@RicardoFreire I don't understand your comment, since $g_3(w_5)$ isn't a sequence.
– Andreas Blass
Nov 30 at 12:19
@RicardoFreire I don't understand your comment, since $g_3(w_5)$ isn't a sequence.
– Andreas Blass
Nov 30 at 12:19
add a comment |
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