Need Help With Using Sequences and Series Test For Convergence/Divergence

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rayroshi

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I am confused about something which seems to be counterintuitive.

According to what I find on the internet and in books, the infinite series (5n+3)/(7n-4), as n approaches infinity, diverges. However, when I try to apply the following line of reasoning, it appears to me that this series converges. Obviously, I am thinking incorrectly, somewhere, but I just can't see where:

The infinite series (5n+3)/(7n-4) can be simplified by multiplying both the numerator and the denominator by 1/n, which then simplifies to (5+3/n)/(7-4/n). So, as n approaches infinity, the terms 3/n and 4/n become 0, which means the series approaches 5/7, a specific number. When a series, in the limit, approaches a specific number, then that series converges.

However, as stated above, according to all of the sources that I can find, this series actually diverges. How can the series that, in the limit, approaches 5/7, a specific number, diverge?

This makes me feel as though the limit test for convergence/divergence can't really be trusted.

Where am I being wrongheaded about this?

Please try to explain it in a way that a non-mathematics major can understand, as I'm 83 years old and trying to learn calculus on my own, using books, YouTube, and now ChatGPt. I don't have an class or instructor to rely upon.

Any help would be greatly appreciated!
 
rayroshi said:
I am confused about something which seems to be counterintuitive.

According to what I find on the internet and in books, the infinite series (5n+3)/(7n-4), as n approaches infinity, diverges. However, when I try to apply the following line of reasoning, it appears to me that this series converges. Obviously, I am thinking incorrectly, somewhere, but I just can't see where:

The infinite series (5n+3)/(7n-4) can be simplified by multiplying both the numerator and the denominator by 1/n, which then simplifies to (5+3/n)/(7-4/n). So, as n approaches infinity, the terms 3/n and 4/n become 0, which means the series approaches 5/7, a specific number. When a series, in the limit, approaches a specific number, then that series converges.

However, as stated above, according to all of the sources that I can find, this series actually diverges. How can the series that, in the limit, approaches 5/7, a specific number, diverge?

This makes me feel as though the limit test for convergence/divergence can't really be trusted.

Where am I being wrongheaded about this?

Please try to explain it in a way that a non-mathematics major can understand, as I'm 83 years old and trying to learn calculus on my own, using books, YouTube, and now ChatGPt. I don't have an class or instructor to rely upon.

Any help would be greatly appreciated!
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You appear to be confusing the terms "series" and "sequence".

The sequence [imath]u_n=\frac{5n+3}{7n-4}[/imath] converges to [imath]\frac{5}{7}[/imath], exactly as you say.

But the series [imath]\sum_{n=1}^\infty\frac{5n+3}{7n-4}[/imath] diverges, because the terms do not converge to zero.

A series is a sum of terms, not just the terms themselves.
 
Suppose we have this infinite series:

[imath]\displaystyle \sum_{n=1}^{\infty} a_n[/imath]

First rule in infinite series:

if [imath]\displaystyle \lim_{n\rightarrow \infty} a_n \neq 0[/imath], the infinite series [imath]\displaystyle \sum_{n=1}^{\infty} a_n[/imath] diverges.
 
Dr.Peterson said:
You appear to be confusing the terms "series" and "sequence".

The sequence [imath]u_n=\frac{5n+3}{7n-4}[/imath] converges to [imath]\frac{5}{7}[/imath], exactly as you say.

But the series [imath]\sum_{n=1}^\infty\frac{5n+3}{7n-4}[/imath] diverges, because the terms do not converge to zero.

A series is a sum of terms, not just the terms themselves.
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Thank you for your response.

No, I am well aware of the distinction between a sequence and a series, which is a basic distinction: 1,2,3... is sequence, but 1+2+3 is a series; pretty obvious stuff; however, what I still don't understand is how a sequence can converge while it's series will, all the while, diverge. I am searching for a more in-depth, fundamental explanation.

This problem reminds me of the example of taking a one-square-inch piece of paper and folding it in half, then folding that in half again...an infinite number of times...then realizing that those infinite number of 'halves' must add up to the original one-square-inch of area, even though an infinite number of areas should result in an infinite amount of area. Crazy stuff. Infinities give me a headache.

Is it some sort of parallel to Zeno's Paradox?; i.e., In my original example problem, although the individual terms, 3/n and 4/n, are becoming progressively smaller an n gets progressively larger, they approach zero but still count for something, as it were, which would then make the series diverge? That's the only thing that would seem to make sense to me.
 
rayroshi said:
what I still don't understand is how a sequence can converge while it's series will, all the while, diverge.
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Consider this series.
[math]1+ \dfrac{1}{2} + \dfrac{1}{2}+ \dfrac{1}{3}+ \dfrac{1}{3}+ \dfrac{1}{3}+ \dfrac{1}{4}+ \dfrac{1}{4}+ \dfrac{1}{4}+ \dfrac{1}{4}...[/math]
This sequence converges to [imath]0.[/imath] However, the series diverges.
[math]1+ \left(\dfrac{1}{2} + \dfrac{1}{2}\right)+ \left(\dfrac{1}{3}+ \dfrac{1}{3}+ \dfrac{1}{3}\right)+ \left(\dfrac{1}{4}+ \dfrac{1}{4}+ \dfrac{1}{4}+ \dfrac{1}{4}\right)...= 1 + 1 + 1 + 1 + ....[/math]
 
blamocur said:
Why? Aren't the definitions fundamental enough here?
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Actually, no. Just learning definitions is definitely not enough, if you mean just memorizing the definitions. Memorizing is not 'fundamental' (as you say) learning, but rather mindless parroting. If I learn the reasons for the definitions, that's a different story. I want to learn the reasons for the definitions. Understanding is a higher-order thinking skill than memorizing, of course.
 
khansaheb said:
I suppose the explanations were not esoteric enough !?!
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'Esoteric' has the connotation of something that relates to an elite, small group that has understanding which is not widely known, and since almost all high schools now offer calculus, that term hardly applies here. I just want to understand the reasons behind a concept; is that so wrong?
 
rayroshi said:
'Esoteric' has the connotation of something that relates to an elite, small group that has understanding which is not widely known, and since almost all high schools now offer calculus, that term hardly applies here. I just want to understand the reasons behind a concept; is that so wrong?
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By the way, do you have an answer to the question I posted? Probably not, I would guess.
 
BigBeachBanana said:
Consider this series.
[math]1+ \dfrac{1}{2} + \dfrac{1}{2}+ \dfrac{1}{3}+ \dfrac{1}{3}+ \dfrac{1}{3}+ \dfrac{1}{4}+ \dfrac{1}{4}+ \dfrac{1}{4}+ \dfrac{1}{4}...[/math]
This sequence converges to [imath]0.[/imath] However, the series diverges.
[math]1+ \left(\dfrac{1}{2} + \dfrac{1}{2}\right)+ \left(\dfrac{1}{3}+ \dfrac{1}{3}+ \dfrac{1}{3}\right)+ \left(\dfrac{1}{4}+ \dfrac{1}{4}+ \dfrac{1}{4}+ \dfrac{1}{4}\right)...= 1 + 1 + 1 + 1 + ....[/math]
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Thanks for your reply. I really appreciate your time and effort.

I can see what you're saying with that excellent way of putting it. It certainly is the best way of showing how an infinite number of increasingly smaller things can diverge that I have ever seen. Now if I can just wrap my head around how such a thing that you have clearly presented to be true can be true, when the pieces/numbers are heading toward zero.... It's all so crazy.

Thanks again for that graphic bit of unassailable logic.
 
Both you and Dr. Peterson have made the good point that the sequence does converge, while the series simultaneously diverges, and while I understand that a sequence is just a list of numbers based on a rule and a series is the summation of those numbers, it seems as though I'm still missing something about the connection/relationship between the two. I guess that, as blamcour said, maybe I'm just stuck with definitions and will have to let it go at that; however, it seems like that is kinda like tossing in the towel and, somehow, just 'moving on' without establishing a real understanding of what's really going on. It must be nice to be like a Euler, Ramanujan, or Gauss who can intuit math.
 
rayroshi said:
Both you and Dr. Peterson have made the good point that the sequence does converge, while the series simultaneously diverges, and while I understand that a sequence is just a list of numbers based on a rule and a series is the summation of those numbers, it seems as though I'm still missing something about the connection/relationship between the two.
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I wish I could tell what you're missing. You should be able to develop some level of intuition about this from enough exposure to examples, though anything involving infinity ultimately has to be left not fully explained, because, after all, we ourselves are finite. A big part of this is accepting that infinite things don't always behave like finite things.

Can you be any more specific about what parts of the relationship don't make sense to you? Is this still what you find unclear?
rayroshi said:
what I still don't understand is how a sequence can converge while it's series will, all the while, diverge. I am searching for a more in-depth, fundamental explanation.

This problem reminds me of the example of taking a one-square-inch piece of paper and folding it in half, then folding that in half again...an infinite number of times...then realizing that those infinite number of 'halves' must add up to the original one-square-inch of area, even though an infinite number of areas should result in an infinite amount of area. Crazy stuff. Infinities give me a headache.

Is it some sort of parallel to Zeno's Paradox?; i.e., In my original example problem, although the individual terms, 3/n and 4/n, are becoming progressively smaller an n gets progressively larger, they approach zero but still count for something, as it were, which would then make the series diverge? That's the only thing that would seem to make sense to me.
Click to expand...
(The first paragraph, your initial question, is the opposite issue from the second, which is essentially Zeno.)
 
rayroshi said:
I'm still missing something about the connection/relationship between the two
Click to expand...
How about: if the series converge then the underlying sequence converges to 0. Otherwise I'll second @Dr.Peterson's suggestion of developing intuition by looking into examples and doing exercises. Wouldn't be surprised if Euler, Ramanujan and Gauss had similar suggestions :)
 
Dr.Peterson said:
I wish I could tell what you're missing. You should be able to develop some level of intuition about this from enough exposure to examples, though anything involving infinity ultimately has to be left not fully explained, because, after all, we ourselves are finite. A big part of this is accepting that infinite things don't always behave like finite things.

Can you be any more specific about what parts of the relationship don't make sense to you? Is this still what you find unclear?

(The first paragraph, your initial question, is the opposite issue from the second, which is essentially Zeno.)
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Thanks again for your reply.

I also wish that I could tell you what I'm missing; however, unfortunately, I have a feeling that what I'm missing would normally lie somewhere in my frontal cerebral cortex, lol!

I have a feeling that you are spot on with you comment about anything involving infinity having to be left not fully explained. It's crazy stuff, full of mathematical paradoxes from what little I have read. It reminds me of the quantum weirdness of cubits being in more than one hazy state of probability at the same time until they are observed. And yes, I sure do agree about what you said re. us being finite. It's all so humbling.

Is the idea of repeatedly folding a one-square-inch piece of paper in half an infinite number of times (clearly impossible, but just for illustration) an example of a diverging sequence? I suppose it would be something like 1, 1/2, 1/4, 1/8... If so, then how could the related series 1 + 1/2 + 1/4 + 1/8... add to more than the original one square inch? I guess that example would be the best way of telling you what I'm missing.
 
rayroshi said:
Is the idea of repeatedly folding a one-square-inch piece of paper in half an infinite number of times (clearly impossible, but just for illustration) an example of a diverging sequence? I suppose it would be something like 1, 1/2, 1/4, 1/8... If so, then how could the related series 1 + 1/2 + 1/4 + 1/8... add to more than the original one square inch? I guess that example would be the best way of telling you what I'm missing.
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No, that is a series whose sequence of terms converges to zero, and whose sum converges to 2. (Obviously it adds up to more than 1; I'm sure you didn't write what you meant.)

But that's not related to folding a piece of paper, which you described here:
rayroshi said:
This problem reminds me of the example of taking a one-square-inch piece of paper and folding it in half, then folding that in half again...an infinite number of times...then realizing that those infinite number of 'halves' must add up to the original one-square-inch of area, even though an infinite number of areas should result in an infinite amount of area. Crazy stuff. Infinities give me a headache.
Click to expand...
The results of folding don't add up; why would they? The sequence you list is just the successive things you'd see as you fold, not pieces that add up.

Infinity will only come close to making sense if you think very carefully, and don't make leaps without thinking. Never run too fast toward infinity (or anything else); you might trip.

But if you cut the square in half, and then cut one half in half, and so on, then you would get 1/2 + 1/4 + 1/8 + ... = 1.

That series is easy to visualize without having to actually imagine taking infinite time. Here is an image from a site worth looking at:

1713055606999.png
That's your series, and it doesn't really take any words to convince people that the series adds up to 1.

Often all it takes is looking at a problem in a different way,
 
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