10
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 ...
- 3,258
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 $n \ln n$ (...
- 6,332
3
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 ...
- 401
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) ...
- 21
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 ...
- 149
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