Mixing
What is the notation for alternating series with “$cdots$”?
$begingroup$
I know that the notation isn't very important, but I am curious. If I have the following alternating series:
$$
sum_{n=0}^{infty} frac{(-1)^n}{n!}
$$
Which of the following notations is the most correct (or the best) for represent the previous series?
begin{align}
&frac{1}{0!}-frac{1}{1!}+frac{1}{2!}-frac{1}{3!}+frac{1}{4!}+cdots tag{1} \[6pt]
&frac{1}{0!}-frac{1}{1!}+frac{1}{2!}-frac{1}{3!}+frac{1}{4!}-cdots tag{2} \[6pt]
&frac{1}{0!}-frac{1}{1!}+frac{1}{2!}-frac{1}{3!}+frac{1}{4!}pmcdots tag{3} \[6pt]
&frac{1}{0!}-frac{1}{1!}+frac{1}{2!}-frac{1}{3!}+frac{1}{4!}+-cdots tag{4} \[6pt]
&frac{1}{0!}-frac{1}{1!}+frac{1}{2!}-frac{1}{3!}+frac{1}{4!}cdots tag{5} \
end{align}
I think that is the equation $(3)$, but I am not sure.
sequences-and-series notation
edited Feb 3 at 15:01
David C. Ullrich
61.8k44095
asked Feb 3 at 5:14
El boritoEl borito
664216
$endgroup$
|
show 6 more comments
$begingroup$
I know that the notation isn't very important, but I am curious. If I have the following alternating series:
$$
sum_{n=0}^{infty} frac{(-1)^n}{n!}
$$
Which of the following notations is the most correct (or the best) for represent the previous series?
begin{align}
&frac{1}{0!}-frac{1}{1!}+frac{1}{2!}-frac{1}{3!}+frac{1}{4!}+cdots tag{1} \[6pt]
&frac{1}{0!}-frac{1}{1!}+frac{1}{2!}-frac{1}{3!}+frac{1}{4!}-cdots tag{2} \[6pt]
&frac{1}{0!}-frac{1}{1!}+frac{1}{2!}-frac{1}{3!}+frac{1}{4!}pmcdots tag{3} \[6pt]
&frac{1}{0!}-frac{1}{1!}+frac{1}{2!}-frac{1}{3!}+frac{1}{4!}+-cdots tag{4} \[6pt]
&frac{1}{0!}-frac{1}{1!}+frac{1}{2!}-frac{1}{3!}+frac{1}{4!}cdots tag{5} \
end{align}
I think that is the equation $(3)$, but I am not sure.
sequences-and-series notation
edited Feb 3 at 15:01
David C. Ullrich
61.8k44095
asked Feb 3 at 5:14
El boritoEl borito
664216
$endgroup$
10
$begingroup$
The best is $sum_{n=0}^{infty} frac{(-1)^n}{n!}$. But of the rest, I'd prefer (2).
$endgroup$
– Lord Shark the Unknown
Feb 3 at 5:16
4
$begingroup$
I also prefer (2). I've never seen (3) or (4).
$endgroup$
– parsiad
Feb 3 at 5:16
4
$begingroup$
2 is best. Also the singular of "series" is oddly enough "series" not "serie" which isn't a word.
$endgroup$
– user608030
Feb 3 at 5:17
4
$begingroup$
I second the above comments. Ellipses are for hiding part of a sentence, so I would also go for (2).
$endgroup$
– Sangchul Lee
Feb 3 at 5:19
3
$begingroup$
I go for $(2)$!
$endgroup$
– manooooh
Feb 3 at 5:19
|
show 6 more comments
$begingroup$
I know that the notation isn't very important, but I am curious. If I have the following alternating series:
$$
sum_{n=0}^{infty} frac{(-1)^n}{n!}
$$
Which of the following notations is the most correct (or the best) for represent the previous series?
begin{align}
&frac{1}{0!}-frac{1}{1!}+frac{1}{2!}-frac{1}{3!}+frac{1}{4!}+cdots tag{1} \[6pt]
&frac{1}{0!}-frac{1}{1!}+frac{1}{2!}-frac{1}{3!}+frac{1}{4!}-cdots tag{2} \[6pt]
&frac{1}{0!}-frac{1}{1!}+frac{1}{2!}-frac{1}{3!}+frac{1}{4!}pmcdots tag{3} \[6pt]
&frac{1}{0!}-frac{1}{1!}+frac{1}{2!}-frac{1}{3!}+frac{1}{4!}+-cdots tag{4} \[6pt]
&frac{1}{0!}-frac{1}{1!}+frac{1}{2!}-frac{1}{3!}+frac{1}{4!}cdots tag{5} \
end{align}
I think that is the equation $(3)$, but I am not sure.
sequences-and-series notation
edited Feb 3 at 15:01
David C. Ullrich
61.8k44095
asked Feb 3 at 5:14
El boritoEl borito
664216
$endgroup$
I know that the notation isn't very important, but I am curious. If I have the following alternating series:
$$
sum_{n=0}^{infty} frac{(-1)^n}{n!}
$$
Which of the following notations is the most correct (or the best) for represent the previous series?
begin{align}
&frac{1}{0!}-frac{1}{1!}+frac{1}{2!}-frac{1}{3!}+frac{1}{4!}+cdots tag{1} \[6pt]
&frac{1}{0!}-frac{1}{1!}+frac{1}{2!}-frac{1}{3!}+frac{1}{4!}-cdots tag{2} \[6pt]
&frac{1}{0!}-frac{1}{1!}+frac{1}{2!}-frac{1}{3!}+frac{1}{4!}pmcdots tag{3} \[6pt]
&frac{1}{0!}-frac{1}{1!}+frac{1}{2!}-frac{1}{3!}+frac{1}{4!}+-cdots tag{4} \[6pt]
&frac{1}{0!}-frac{1}{1!}+frac{1}{2!}-frac{1}{3!}+frac{1}{4!}cdots tag{5} \
end{align}
I think that is the equation $(3)$, but I am not sure.
sequences-and-series notation
sequences-and-series notation
edited Feb 3 at 15:01
David C. Ullrich
61.8k44095
asked Feb 3 at 5:14
El boritoEl borito
664216
edited Feb 3 at 15:01
David C. Ullrich
61.8k44095
asked Feb 3 at 5:14
El boritoEl borito
664216
edited Feb 3 at 15:01
David C. Ullrich
61.8k44095
edited Feb 3 at 15:01
David C. Ullrich
61.8k44095
edited Feb 3 at 15:01
David C. Ullrich
61.8k44095
61.8k44095
asked Feb 3 at 5:14
El boritoEl borito
664216
asked Feb 3 at 5:14
El boritoEl borito
664216
asked Feb 3 at 5:14
El boritoEl borito
664216
664216
10
$begingroup$
The best is $sum_{n=0}^{infty} frac{(-1)^n}{n!}$. But of the rest, I'd prefer (2).
$endgroup$
– Lord Shark the Unknown
Feb 3 at 5:16
4
$begingroup$
I also prefer (2). I've never seen (3) or (4).
$endgroup$
– parsiad
Feb 3 at 5:16
4
$begingroup$
2 is best. Also the singular of "series" is oddly enough "series" not "serie" which isn't a word.
$endgroup$
– user608030
Feb 3 at 5:17
4
$begingroup$
I second the above comments. Ellipses are for hiding part of a sentence, so I would also go for (2).
$endgroup$
– Sangchul Lee
Feb 3 at 5:19
3
$begingroup$
I go for $(2)$!
$endgroup$
– manooooh
Feb 3 at 5:19
|
show 6 more comments
10
$begingroup$
The best is $sum_{n=0}^{infty} frac{(-1)^n}{n!}$. But of the rest, I'd prefer (2).
$endgroup$
– Lord Shark the Unknown
Feb 3 at 5:16
4
$begingroup$
I also prefer (2). I've never seen (3) or (4).
$endgroup$
– parsiad
Feb 3 at 5:16
4
$begingroup$
2 is best. Also the singular of "series" is oddly enough "series" not "serie" which isn't a word.
$endgroup$
– user608030
Feb 3 at 5:17
4
$begingroup$
I second the above comments. Ellipses are for hiding part of a sentence, so I would also go for (2).
$endgroup$
– Sangchul Lee
Feb 3 at 5:19
3
$begingroup$
I go for $(2)$!
$endgroup$
– manooooh
Feb 3 at 5:19
10
10
$begingroup$
The best is $sum_{n=0}^{infty} frac{(-1)^n}{n!}$. But of the rest, I'd prefer (2).
$endgroup$
– Lord Shark the Unknown
Feb 3 at 5:16
$begingroup$
The best is $sum_{n=0}^{infty} frac{(-1)^n}{n!}$. But of the rest, I'd prefer (2).
$endgroup$
– Lord Shark the Unknown
Feb 3 at 5:16
4
4
$begingroup$
I also prefer (2). I've never seen (3) or (4).
$endgroup$
– parsiad
Feb 3 at 5:16
$begingroup$
I also prefer (2). I've never seen (3) or (4).
$endgroup$
– parsiad
Feb 3 at 5:16
4
4
$begingroup$
2 is best. Also the singular of "series" is oddly enough "series" not "serie" which isn't a word.
$endgroup$
– user608030
Feb 3 at 5:17
$begingroup$
2 is best. Also the singular of "series" is oddly enough "series" not "serie" which isn't a word.
$endgroup$
– user608030
Feb 3 at 5:17
4
4
$begingroup$
I second the above comments. Ellipses are for hiding part of a sentence, so I would also go for (2).
$endgroup$
– Sangchul Lee
Feb 3 at 5:19
$begingroup$
I second the above comments. Ellipses are for hiding part of a sentence, so I would also go for (2).
$endgroup$
– Sangchul Lee
Feb 3 at 5:19
3
3
$begingroup$
I go for $(2)$!
$endgroup$
– manooooh
Feb 3 at 5:19
$begingroup$
I go for $(2)$!
$endgroup$
– manooooh
Feb 3 at 5:19
|
show 6 more comments
2 Answers
2
active
oldest
votes
$begingroup$
Converting a comment to an answer ...
(2) is most common. (See, for instance, some examples in Wikipedia's "Taylor series" entry.) So long as you've included enough terms to establish the pattern, you're fine. In cases where the pattern may not be entirely clear, the sigma notation is best; alternatively, you can provide this hybrid form
$$frac{1}{0!}−frac{1}{1!}+frac{1}{2!}−frac{1}{3!}+frac{1}{4!}−cdots+frac{(−1)^n}{n!}+cdots$$
answered Feb 3 at 5:52
BlueBlue
49.7k870158
$endgroup$
add a comment |
$begingroup$
I would use $(5)$ or maybe $(2)$, but I don't think it is very important. What is important is to show enough terms that the pattern is obvious. I would not rely on the trailing sign to convey that information, which is why I would use $(5)$. I think $(5)$ is sufficient, but I might show the next term before I went to dots.
answered Feb 3 at 5:30
Ross MillikanRoss Millikan
301k24200375
$endgroup$
1
$begingroup$
+1 If understanding the pattern comes down to the character before trailing ellipsis, then you either haven't demonstrated the pattern well enough, or (more likely) you should find a more precise way to express it.
$endgroup$
– Theo Bendit
Feb 3 at 5:45
$begingroup$
This is a very good answer, thank you for your knowledge
$endgroup$
– El borito
Feb 4 at 4:54
add a comment |
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2 Answers
2
active
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2 Answers
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active
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$begingroup$
Converting a comment to an answer ...
(2) is most common. (See, for instance, some examples in Wikipedia's "Taylor series" entry.) So long as you've included enough terms to establish the pattern, you're fine. In cases where the pattern may not be entirely clear, the sigma notation is best; alternatively, you can provide this hybrid form
$$frac{1}{0!}−frac{1}{1!}+frac{1}{2!}−frac{1}{3!}+frac{1}{4!}−cdots+frac{(−1)^n}{n!}+cdots$$
answered Feb 3 at 5:52
BlueBlue
49.7k870158
$endgroup$
add a comment |
$begingroup$
Converting a comment to an answer ...
(2) is most common. (See, for instance, some examples in Wikipedia's "Taylor series" entry.) So long as you've included enough terms to establish the pattern, you're fine. In cases where the pattern may not be entirely clear, the sigma notation is best; alternatively, you can provide this hybrid form
$$frac{1}{0!}−frac{1}{1!}+frac{1}{2!}−frac{1}{3!}+frac{1}{4!}−cdots+frac{(−1)^n}{n!}+cdots$$
answered Feb 3 at 5:52
BlueBlue
49.7k870158
$endgroup$
add a comment |
$begingroup$
Converting a comment to an answer ...
(2) is most common. (See, for instance, some examples in Wikipedia's "Taylor series" entry.) So long as you've included enough terms to establish the pattern, you're fine. In cases where the pattern may not be entirely clear, the sigma notation is best; alternatively, you can provide this hybrid form
$$frac{1}{0!}−frac{1}{1!}+frac{1}{2!}−frac{1}{3!}+frac{1}{4!}−cdots+frac{(−1)^n}{n!}+cdots$$
answered Feb 3 at 5:52
BlueBlue
49.7k870158
$endgroup$
Converting a comment to an answer ...
(2) is most common. (See, for instance, some examples in Wikipedia's "Taylor series" entry.) So long as you've included enough terms to establish the pattern, you're fine. In cases where the pattern may not be entirely clear, the sigma notation is best; alternatively, you can provide this hybrid form
$$frac{1}{0!}−frac{1}{1!}+frac{1}{2!}−frac{1}{3!}+frac{1}{4!}−cdots+frac{(−1)^n}{n!}+cdots$$
answered Feb 3 at 5:52
BlueBlue
49.7k870158
answered Feb 3 at 5:52
BlueBlue
49.7k870158
answered Feb 3 at 5:52
BlueBlue
49.7k870158
answered Feb 3 at 5:52
BlueBlue
49.7k870158
49.7k870158
add a comment |
add a comment |
$begingroup$
I would use $(5)$ or maybe $(2)$, but I don't think it is very important. What is important is to show enough terms that the pattern is obvious. I would not rely on the trailing sign to convey that information, which is why I would use $(5)$. I think $(5)$ is sufficient, but I might show the next term before I went to dots.
answered Feb 3 at 5:30
Ross MillikanRoss Millikan
301k24200375
$endgroup$
1
$begingroup$
+1 If understanding the pattern comes down to the character before trailing ellipsis, then you either haven't demonstrated the pattern well enough, or (more likely) you should find a more precise way to express it.
$endgroup$
– Theo Bendit
Feb 3 at 5:45
$begingroup$
This is a very good answer, thank you for your knowledge
$endgroup$
– El borito
Feb 4 at 4:54
add a comment |
$begingroup$
I would use $(5)$ or maybe $(2)$, but I don't think it is very important. What is important is to show enough terms that the pattern is obvious. I would not rely on the trailing sign to convey that information, which is why I would use $(5)$. I think $(5)$ is sufficient, but I might show the next term before I went to dots.
answered Feb 3 at 5:30
Ross MillikanRoss Millikan
301k24200375
$endgroup$
1
$begingroup$
+1 If understanding the pattern comes down to the character before trailing ellipsis, then you either haven't demonstrated the pattern well enough, or (more likely) you should find a more precise way to express it.
$endgroup$
– Theo Bendit
Feb 3 at 5:45
$begingroup$
This is a very good answer, thank you for your knowledge
$endgroup$
– El borito
Feb 4 at 4:54
add a comment |
$begingroup$
I would use $(5)$ or maybe $(2)$, but I don't think it is very important. What is important is to show enough terms that the pattern is obvious. I would not rely on the trailing sign to convey that information, which is why I would use $(5)$. I think $(5)$ is sufficient, but I might show the next term before I went to dots.
answered Feb 3 at 5:30
Ross MillikanRoss Millikan
301k24200375
$endgroup$
I would use $(5)$ or maybe $(2)$, but I don't think it is very important. What is important is to show enough terms that the pattern is obvious. I would not rely on the trailing sign to convey that information, which is why I would use $(5)$. I think $(5)$ is sufficient, but I might show the next term before I went to dots.
answered Feb 3 at 5:30
Ross MillikanRoss Millikan
301k24200375
answered Feb 3 at 5:30
Ross MillikanRoss Millikan
301k24200375
answered Feb 3 at 5:30
Ross MillikanRoss Millikan
301k24200375
answered Feb 3 at 5:30
Ross MillikanRoss Millikan
301k24200375
301k24200375
1
$begingroup$
+1 If understanding the pattern comes down to the character before trailing ellipsis, then you either haven't demonstrated the pattern well enough, or (more likely) you should find a more precise way to express it.
$endgroup$
– Theo Bendit
Feb 3 at 5:45
$begingroup$
This is a very good answer, thank you for your knowledge
$endgroup$
– El borito
Feb 4 at 4:54
add a comment |
1
$begingroup$
+1 If understanding the pattern comes down to the character before trailing ellipsis, then you either haven't demonstrated the pattern well enough, or (more likely) you should find a more precise way to express it.
$endgroup$
– Theo Bendit
Feb 3 at 5:45
$begingroup$
This is a very good answer, thank you for your knowledge
$endgroup$
– El borito
Feb 4 at 4:54
1
1
$begingroup$
+1 If understanding the pattern comes down to the character before trailing ellipsis, then you either haven't demonstrated the pattern well enough, or (more likely) you should find a more precise way to express it.
$endgroup$
– Theo Bendit
Feb 3 at 5:45
$begingroup$
+1 If understanding the pattern comes down to the character before trailing ellipsis, then you either haven't demonstrated the pattern well enough, or (more likely) you should find a more precise way to express it.
$endgroup$
– Theo Bendit
Feb 3 at 5:45
$begingroup$
This is a very good answer, thank you for your knowledge
$endgroup$
– El borito
Feb 4 at 4:54
$begingroup$
This is a very good answer, thank you for your knowledge
$endgroup$
– El borito
Feb 4 at 4:54
add a comment |
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10
$begingroup$
The best is $sum_{n=0}^{infty} frac{(-1)^n}{n!}$. But of the rest, I'd prefer (2).
$endgroup$
– Lord Shark the Unknown
Feb 3 at 5:16
4
$begingroup$
I also prefer (2). I've never seen (3) or (4).
$endgroup$
– parsiad
Feb 3 at 5:16
4
$begingroup$
2 is best. Also the singular of "series" is oddly enough "series" not "serie" which isn't a word.
$endgroup$
– user608030
Feb 3 at 5:17
4
$begingroup$
I second the above comments. Ellipses are for hiding part of a sentence, so I would also go for (2).
$endgroup$
– Sangchul Lee
Feb 3 at 5:19
3
$begingroup$
I go for $(2)$!
$endgroup$
– manooooh
Feb 3 at 5:19