# Linked Questions

7answers
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### Why do we say the harmonic series is divergent? [duplicate]

If we have $\Sigma\frac{1}{n}$, why do we say it is divergent? Yes, it is constantly increasing, but after a certain point, $n$ will be so large that we will be certain of millions of digits. If we ...
2answers
796 views

### Is the sequence of sums of inverse of natural numbers bounded? [duplicate]

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, ...
4answers
475 views

2answers
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### Prove sequence $H_n=1+\frac{1}{2}+\frac{1} {3}+\dots+\frac{1}{n}$ does not converge. [duplicate]

I want to prove that the harmonic series $H_n=1+\frac{1}{2}+\frac{1} {3}+\dots+\frac{1}{n}$ does not converge. My attempt: to see it does not converge, I try to prove that it is unbounded and it is ...
5answers
405 views

I have tried this as : $$p_n=\sum_{k=1}^n\frac{1}{k}=1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n-1}+\frac{1}{n}$$ $$p_{n-1}=\sum_{k=1}^{n-1}\frac{1}{k}=1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n-1}... 3answers 822 views ### Harmonic Series Convergence [duplicate] Could someone explain to me graphically how the harmonic series can be divergent while the alternating harmonic series can be convergent since they are both using 1/n in their series and going towards ... 3answers 216 views ### How do I prove that the series \sum_{n=1}^\infty \frac{1}{n} diverges? [duplicate] I am studying infinite series in class. Our teacher showed us how the series$$\sum_{n=1}^\infty \frac{1}{n}=1+1/2+1/3+...$$diverges because$$\int_{1}^\infty \frac {1}{n}dn=\infty$$Is there a ... 2answers 520 views ### Harmonic Series. [duplicate] I don't know if my proof is correct. Let be \begin{eqnarray} H_n &=& 1 + \dfrac{1}{2} + \dfrac{1}{3} + \dotsb + \dfrac{1}{n} \\ &+& \\ H_n &=& \dfrac{1}{n} + \dfrac{1}{n-1}... 3answers 250 views ### Partial sums of harmonic series [duplicate] I've been given the following problem. Prove that$$\lim_{N\to\infty}\sum_{i=1}^N\frac1i=\infty.$$In other words I need to prove that the partial sum of the harmonic series diverges. I know ... 2answers 105 views ### does \sum_{n=1}^{\infty} \frac{1}{n} converge [duplicate] Does \sum_{n=1}^{\infty} \frac{1}{n} converge? I actually do not know this, and have no steps to prove it, but my logic thinks that it is convergent. But when graphed, it takes ages for the function ... 0answers 293 views ### Why does Σ1/x diverge? [duplicate] Why does the following series diverge?$$\sum_{n = 1}^\infty\frac{1}{n}$$I've tried to make sense of it, but can't seem to wrap my head around it. Thanks. 3answers 219 views ### Showing that X_n = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} is not bounded above. [duplicate] Possible Duplicate: Why does the series \frac 1 1 + \frac 12 + \frac 13 + \cdots not converge? Prove that the sequence converges I have to show that X_n is not bounded above,$$0<1\le1\$...

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