Bad solution in a problem with closed discussion

The most upvoted solution in https://brilliant.org/practice/calculus-warmups-level-2-challenges/?p=6 uses some mathematical wrong methods. He splits a converging sum in 2 diverging sums, makes a index shift and subtracted $\infty$ from $\infty$ to get a final solution. Because discussions are closed, there is no possibility to warn, that this method for solution is wrong. Can a moderator help here?

#Calculus

Note by Stephan Hensel
2 years, 9 months ago

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Comments

I believe that you are referring to the 5th problem of that set. You could notify the solution writer here on this note using @ followed by their name and ask them to clarify their solution, (if they feel that they need to), but I suspect that the option to edit solutions is closed too, so any changes would have to be made directly by staff (and not just a moderator, (such as me)). The solution writer in question is excellent and it is not a "bad solution" but I do see your point about the splitting into diverging series; perhaps there is a more rigorous way of representing the telescoping sum, such as

1+lim⁡N→∞∑n=1N(1n−1n+1)=1+lim⁡N→∞(∑n=1N1n−∑n=1N1n+1)=1+1+lim⁡N→∞(∑n=2N1n−∑n=2N+11n)=2−lim⁡N→∞1N+1=21 + \displaystyle \lim_{N \to \infty} \sum_{n=1}^{N} \left(\dfrac{1}{n} - \dfrac{1}{n + 1}\right) = 1 + \lim_{N \to \infty} \left(\sum_{n=1}^{N} \dfrac{1}{n} - \sum_{n=1}^{N} \dfrac{1}{n + 1}\right) = 1 + 1 + \lim_{N \to \infty} \left( \sum_{n = 2}^{N} \dfrac{1}{n} - \sum_{n = 2}^{N + 1} \dfrac{1}{n} \right) = 2 - \lim_{N \to \infty} \dfrac{1}{N + 1} = 2.

Brian Charlesworth - 2 years, 9 months ago
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