Timeline for How to deal with "Why can't I just do ......" in real analysis?
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 14, 2019 at 16:43 | comment | added | timtfj | I think the student is basically saying "$\sum a_n$ converges to $A$, $\sum b_n$ diverges to $\infty$, therefore $\sum (a_n+b_n)$ diverges to $A+\infty$". | |
| Dec 7, 2018 at 17:03 | comment | added | SBK | I think you've rephrased what I said that the student said. I deliberately wrote that the student used the displayed equation in my question. So, asked to show that $\sum_n (a_n + b_n)$ diverges and you start by saying: "First write $\sum_{n=1}^{\infty}(a_n + b_n) = \sum_{n=1}^{\infty}a_n + \sum_{n=1}^{\infty}b_n$... and then...."etc. The first step is not OK because you've re-ordered an infinite series without knowing anything about its convergence (yet). | |
| Dec 7, 2018 at 13:21 | comment | added | Jasper | Perhaps that splitting up the sum is not justified unless proven correct for the given sequences. | |
| Dec 7, 2018 at 13:17 | comment | added | ruferd | I believe the issue is that for the instructor, this is obviously wrong because the student hasn't proven/disproven the claim using strict definitions, like one usually does in analysis. Instead, the student has used an argument that is true most of the time without considering these tiny corner cases where his argument is not actually true. (Similar to saying, "I have a continuous function, so let me take the derivative of it" without realizing that continuous doesn't always imply differentiable, such as absolute value not being differentiable at $0$. | |
| Dec 7, 2018 at 9:49 | history | answered | Dominique | CC BY-SA 4.0 |