Extremely helpful. The hallmark of a great mathematician is to convey the magic of the subject as smoothly and in as simplified manner as possible. You did that job perfectly.
Whenever I need to understand details bit by bit, I chose Krista King. Math couldn't get any easier the way she explains it. Thanks for doing what you the best.
This video explained everything. this is really great. I didn't even knew Monotone sequence. but with this explanation,now Ik both Bounded and monotone sequence and how to know that sequence is which one.
That's the tutors job I guess (actually teaching). Professors are just there to present a slide show presentation every class which students forget 90% of.
I would want my students to do more than just look at the first few terms to prove that a series is increasing/decreasing by either using the derivative of the function f(x) such that a_n = f(n) or by showing that a_n+1< a_n if decreasing and vice versa if increasing. If a student only showed that the function's first few terms had that trend, they could be tricked by one of those series that increased at first and then decreased or vice versa. Check this series out: a_n = (n-5)^2/10 + 10/n. Your method would make the student think that the sequence is decreasing monotonically when in fact it switches and starts to increase and has no limit as n -> oo.
+MathBySarah I agree! :) And I do teach that using the derivative to prove the series increases or decreases everywhere is the safest way to go. But sometimes, if it's simple enough, we can tell that the series is increasing or decreasing just by looking at it. And in that case, with these longer problems I'll skip that additional explanation.
You teach it so good... I understand it but I have a question that is What is difference between limit x approach 0 and upper bound............from India 🇮🇳
extremely useful. i pay my professor thousands of dollars and explain shit and all that money doesnt buy me anything other than my pockets getting empty but your videos are great. keep posting. #respect #love
so basically if a sequence is convergent then it is bounded n we can directly write it as bounded but for divergent sequences we have to do the whole brain storming ?
If the sequence was decreasing monotonic, and the first term is the largest value thats its bounded above. Do you then find the value thats it bounded below by finding the limit as n tends to negative infinity? Or n tends to positive infinity?
Unfortunately, you are incorrect in couple of places. First, being bounded does not require monotonicity. As a matter of fact these two concept are separate. You might have a mono tonic sequence that is bounded or unbounded. A sequence could be bounded and not mono tonic, for example {sin(n)} is bounded and not monotonic. Second the graph of a sequence is not a continuous curve, just bunch of points on the plane.
Hello Kristen please answer me Alternating sequence like 1,-1,1,-1, Is it dive but , why ?! And it is monotonic or not ?! Could you give me clear idea about alternating sequence Please help me
what happens if a(N+1) is defined in terms of a(N) and we are not provided the general a(N) sequence except the first term then how will i predict the sequence(a(N)) is convergent or not?
Any converging sequence is bounded. To show that the second sequence is monotonically increasing you could consider the function y(x)=\frac{2x-3}{3x+4} for which y'(x)>0. P.S. Greetings from Russia!;)
I'm not sure I follow your logic. It seems like you suggest that because it is non-monotonic, it cannot possibly be bounded, but I can still imagine a non-monotonic sequence that is bounded above and below... If all the positive values converged on a single point and all the negative values converged on another. It seems to be just a coincidence that there is no upper or lower limit in your example, totally unrelated monotonicity.
