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Following limit appears a bit hard to evaluate to anything reasonable:$$\lim_{n\rightarrow\infty}=\left(1+\sqrt{2}+\sqrt[3]{3}+...+\sqrt[n]{n}\right) \ln\frac{2n+1}{n}$$umm... any ideas, hints?

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As $\sqrt[n]{n} \to 1$, the sum diverges. And $\log\frac{2n+1}n \to \log 2$. –  martini Nov 7 '12 at 9:27
    
what if $ \ln (\frac{2n+1}{n}) = \ln (\frac{n+1}{n})$? –  clark Nov 7 '12 at 9:37
    
@clark: just try it using <your favorite software>. My approaches 1 as far as I let it run... –  Gottfried Helms Nov 7 '12 at 17:53

1 Answer 1

up vote 4 down vote accepted

It’s much easier than it looks. First,

$$\lim_{n\to\infty}\ln\frac{2n+1}n=\lim_{n\to\infty}\left(2+\frac1n\right)=\ln 2\;,$$

so you can ignore this factor and concentrate on

$$\lim_{n\to\infty}\left(1+\sqrt2+\sqrt[3]3+\ldots+\sqrt[n]n\right)\;.$$

Clearly $\sqrt[n]n\ge 1$ for each positive integer $n$, so ... ?

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The lesson Brian is trying to teach you here, seems to be: Look at the individual pieces (factors in this case) before tackling the whole. It's an easy lesson, but hard to learn. (I wish I knew why. I see students all the time who struggle because they haven't got it.) –  Harald Hanche-Olsen Nov 7 '12 at 10:52
    
@Harald: Part of it may be an inability to recognize that the individual pieces can be isolated, but lack of confidence is a large part of it, I think: the student simply feels overwhelmed. The other common manifestation is snatching at the first answer or technique that seems to the student to have some connection with the problem without thinking about whether it makes sense. ‘Whoa. Stop. Now slow down and think about what you just said/wrote. Does X make any sense at all?’ instead of Relax. Just take it one piece/step at a time. –  Brian M. Scott Nov 7 '12 at 16:10
    
I appreciate (and I hope I understand the lesson) your answer, but comments are also very useful. It was my fault not to try hard enough, I will do my best to improve now. Thank you for your time and hints. –  fdhd Nov 7 '12 at 17:48
    
@user46034: You’re welcome. Don’t take the comments too personally: those two reactions are something to watch out for, but Harald and I really were talking about a general phenomenon, not about you in particular. And don’t be too sure that you didn’t try hard enough: sometimes you just get caught in a mental blind alley and don’t see a way out no matter how hard you try. (Leaving the problem for a while and coming back to it for a fresh start sometimes helps.) –  Brian M. Scott Nov 7 '12 at 18:06

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