Monotonic Sequences and Bounded Sequences - Calculus 2
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- Опубліковано 25 бер 2018
- This calculus 2 video tutorial provides a basic introduction into monotonic sequences and bounded sequences. A monotonic sequence is a sequence that is always increasing or decreasing. You can prove that a sequence is always increasing by showing that the next term is greater than the previous term. This video also discusses bounded sequences. A sequence can be bounded above or have an upper bound it it has a maximum value it can reach. If there is a minimum value in the sequence, then it has a lower bound or it's bounded below. A sequence that is bounded above and bounded below is said to be bounded. A monotonic sequence that is bounded is said to be convergent. This video explains how to determine the convergence and divergence of a sequence by determining if its a monotonic sequence and if its a bounded sequence.
Improper Integrals:
• Improper Integrals - C...
Converging & Diverging Sequences:
• Converging and Divergi...
Monotonic & Bounded Sequences:
• Monotonic Sequences an...
Absolute Value Theorem - Sequences:
• Absolute Value Theorem...
Squeeze Theorem - Sequences:
• Squeeze Theorem For Se...
________________________________
Geometric Series & Sequences:
• Geometric Series and G...
Introduction to Series - Convergence:
• Convergence and Diverg...
Divergence Test For Series:
• Divergence Test For Se...
Harmonic Series:
• Harmonic Series
Telescoping Series:
• Telescoping Series
__________________________________
Integral Test For Divergence:
• Calculus 2 - Integral ...
Remainder Estimate - Integral Test:
• Remainder Estimate For...
P-Series:
• P-series
Direct Comparison Test:
• Direct Comparison Test...
Limit Comparison Test:
• Limit Comparison Test
___________________________________
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Professor Organic Chemistry Tutor, thank you for explaining Monotonic Sequences and Bounded Sequences in Calculus Two. I also encountered Monotonic and Bounded Sequences in Advanced Calculus; however, I did not understand Monotonic/Bounded Sequences until I watched and analyzed this great video from start to finish. This is an error free video/lecture on UA-cam TV with the Organic Chemistry Tutor.
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On my calculus textbook by James Stewart, it says that a sub n is less than a sub n+1. But you said a sub n is less than or equal to a sub n+1. Your definition contradicts the textbook. But great video overall
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9:48 why do you need to divide it by A_n+1 for this case, but not for other ones I am still confused as to why this is the case?
Why don't you make a separate video on complete bolzano-weierstrass theorem
1:10 doesn't that make a horizontal line both increasing and decreasing?
It was great
if d(an)/dn is positive can we confirm that a function is monotically increasing?
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I thought, because of the growth rate theorem, n! infinity 3^n/n! would be an infinite series. Thus would make it a monotonic, lower bounded, divergent series
Legendary
Just a clarification,
at 30:51, you said it is convergent when n starts at 2.
But regardless the value of n (whether it is positive or negative), it is no longer convergent, right?
Yes
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Is there a tip/trick for me to keep up with the math needed to understand this better? I find myself understanding things when he specifically explains them (in detail) but If I was supposed to intuitively get it right I just wouldn't have it on my mind. Thanks!
If you're talking about the algebraic tricks he used, such as manipulating inequalities, I couldn't hope to explain it better than @TheOrganicChemistryTutor. However, my main takeaway from the entire video is that a sequence converges when it is bounded and monotonic. A bounded sequence is a sequence that has a floor and ceiling (lower and upper bounds) as n goes to infinity. A monotonic sequence is a sequence that only increases or only decreases in the long run. Note how in the last example, the sequence wasn't monotonic on n >= 1, but it was monotonic on n >= 2.
Is there a sequence that does not bound by up or down?
What about 1/x where the sequence is bounded and is always decreasing monotonic but is divergent? Doesn’t that break the rule?
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15:59 i couldn't understand what he said and I don't know how he did that. can someone help me please?
1/x^2 is the same as x^-2 so derivative of x^-2 is -2x^-3 which is simply -2/x^3
I find the concepts of sequences bounded above and/or bounded below kind of hard to understand because the semantic is kind of counter-intuitive for instance bounded below means that the sequence has a lower bound, but from face value it sounds like the sequence would be below a limit by the words "bounded below".
26:47 isn't this incorrect. by the definition of a monotonic series a_n >= a_n+1 or a_n
I’m also quite confused by this, since I’ve only ever seen that as the definition, when it clearly does not work here
It strictly decreases after the value n = 2.
Why did you add instead of distribute the "2" in the denominator at 5:52
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I have a question: is it true that whenever we want to determine the monotonicity of a sequence {a_n} we could always use the Calculus method of taking derivative of continuous function f(x) with f(n) = a_n? And if f(x) is increasing on an interval then {a_n} is increasing on that interval and vice versa if f(x) is decreasing on an interval then {a_n} is decreasing on that interval?
Is this obvious or is there some theorem/corollary stating this?
If f(x) is differentiable, then this certainly holds
n-> infinite, Lim An = 0, which doesn't guarantee it converges.
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Why should the monotonic sequence at 11:45 be greater than or equal to, shouldn't it just be greater than?
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20:45 that doesn't mean that the sequence converges if the limit found is 0. It's when the function an converges that its limit is 0
so monotonic functions can have constant sections?
Hello,
From my experience it doesn’t appear to be a good strategy to assume that an
You make the assumption based on your observation by plugging in the numbers.
He should have proved it by mathematical induction
9:50 why are we dividing an/an+1, because on the first problem we did not have to divide so I am confused on that.
Because the sequence is decreasing.
17:57 I feel dumb, but what happened to the denominator? Why is it just (n+1)^2 >= n^2?
Cross multiplication