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I'm reading through Spivak Ch.22 (Infinite Sequences) right now. He mentioned in the written portion that it's often not a trivial matter to determine the boundedness of sequences. With that in mind, he gave us a sequence to chew on before we learn more about boundedness. That sequence is:

$$1, 1+\frac{1}{2}, 1+\frac{1}{2}+\frac{1}{3}, 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}, . . .$$

I know that a sequence is bounded above if there is a number $M$ such that $a_n\leq M$ for all $n$. Any hints here?

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marked as duplicate by Arnaud D., Martin R, user159517, Saad, Matthew Towers Jan 10 at 16:24

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    $\begingroup$ It is good that you are attending to the definition of boundedness. So, can you write down a suitable $M$ with $a_n\le M$ for all $n$? $\endgroup$ – Lord Shark the Unknown Jan 10 at 5:08
  • $\begingroup$ @LordSharktheUnknown I'm not too sure what that $M$ would be. Can you give a hint? $\endgroup$ – kyle campbell Jan 10 at 5:14
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    $\begingroup$ Those aren't subsequences. In any case, are you sure you have transcribed Spivak's sequence correctly? $\endgroup$ – Lord Shark the Unknown Jan 10 at 5:17
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    $\begingroup$ It is just divergence of harmonic series. You can search the net for 'divergence of harmonic series'. $\endgroup$ – Kavi Rama Murthy Jan 10 at 5:23
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    $\begingroup$ A classic estimate: for each $n$, consider estimating $a_{2^n}$. $\endgroup$ – xbh Jan 10 at 5:26
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This is a classic example of how our intuition can be wrong concerning infinities. As @Kavi said in the comments, this is the harmonic series and indeed diverges. I believe that Spivak provided this as an example to “chew on” in order to show how non trivial boundedness can be, as on first glance many people believe this sequence should be bounded.

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    $\begingroup$ Indeed! Thank you for that input. $\endgroup$ – kyle campbell Jan 10 at 5:57
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    $\begingroup$ This is not the harmonic series, it is the sequence of partial sums of the harmonic series. A sequence and a series are two different things. Other important sequences linked to a series are the sequence of the terms ($\{1/n\}$ in this case) and the sequence of partial rests. $\endgroup$ – A. Arredondo Jan 10 at 9:16
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Indeed, you can notice that

$$a_1=1,a_4>2(=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{4}), a_8>\frac{5}{2}(=a_4+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}),a_{16}>3(=a_{8}+\frac{1}{16}+\frac{1}{16}+..8times)$$, so the general pattern is $a_{2^{2n}}>n+1$, so given an upper bound M, you can find a natural number n s.t. $n\geq M$ by archimedian property and so $a_{2^{2n}}>n+1>M$ and hence it can't be bounded.

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