11 votes

How can we explain intuitively the convergence and divergence of these two series?

Look at a simpler example first: $(1.000000000001)^n$ compared to $0.9999999999^n$. Do they accept that the first sequence tends to $\infty$ and the second to $0$ even though it would take quite a ...
KCd's user avatar
  • 3,446
6 votes

Overcoming Dyslexia and Building Intuition

This might be an unpopular answer but I'm going to try to be honest and real with you. Math gets hard and unintuitive for everyone at some point, and that point is different for everyone -- for some ...
Justin Skycak's user avatar
6 votes

How can we explain intuitively the convergence and divergence of these two series?

For me, the intuition just comes from the integral test (which is itself intuitive since a series is just a Riemann sum of rectangles with unit width). The $n$th prime is asymptotically $n \ln n$ (...
Justin Skycak's user avatar
4 votes

Are these explanations of variance and covariance intuitive?

Here is an everyday example familiar to many students. The time it takes to walk to school has less variance than the time it takes to get to school by bus (this is true everywhere I've lived). This ...
Dan Fox's user avatar
  • 5,839
4 votes

"Rough subitising / estimation" for better intuition and ability to apply arithmetic

Beyond the visual guesswork of scanning the looseleaf to estimate how many lines are there, the process described amounts to modeling the situation based on estimated values of a few important key ...
Steve's user avatar
  • 1,514
3 votes

Overcoming Dyslexia and Building Intuition

I am a student in mathematics, and I've always been told that mathematics is not for everyone and that our brain is "weird" or works differently so that we can do maths. But as I read maths ...
bml64's user avatar
  • 311
2 votes

For calculus students, what should be the intuition or motivation behind series?

The fact of the matter is series is sort of viscerally off the beaten track of limits/derivatives/minmax apps/integral toolbag/integral apps. It's there because it is needed later (and covered more) ...
guest troll's user avatar
2 votes

How can we explain intuitively the convergence and divergence of these two series?

Intuitively, to me, it means that if you take the positive number line, put a blue dot at every prime, and a red dot on all the the numbers of the form $n^{1.000000000001}$, then eventually, very far ...
Arthur's user avatar
  • 399
1 vote

How can we explain intuitively the convergence and divergence of these two series?

Consider the fact that $\sum_{n=1}^\infty n^x$ converges if $x<0$, diverges if $x>0$. Clearly the transition from just a little bit negative to just a little bit positive makes a big change to ...
Simon Crase's user avatar
1 vote

Are these explanations of variance and covariance intuitive?

I think it's good to try to help students develop some intuition for variance and covariance. However, a pedagogical problem I see with your variance example is that both distributions have the same ...
Justin Hancock's user avatar

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