Linked Questions

1
vote
7answers
19k views

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 ...
8
votes
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, ...
8
votes
4answers
475 views

What are different ways to prove that $\sum_{n=1}^{\infty}\frac 1n$ is divergent? [duplicate]

I have been studying Cauchy criterion for sequences, and have come across a rather simple proof for the harmonic series, and why it diverges. More so, we have the following: $$\sum_{n=1}^{\infty}\...
6
votes
3answers
569 views

How to find the limit of a sum of reciprocals $\lim_{n\to\infty}(1 + \frac{1}{2} + \frac{1}{3} + \cdots+ \frac{1}{n})$? [duplicate]

There's a limit that I am unable to solve. I think it should be equal to $\infty$. $$\lim_{n\to\infty}\left(1 + \frac{1}{2} + \frac{1}{3} + \cdots+ \frac{1}{n}\right)$$
2
votes
2answers
457 views

How can $\sum_{n=1}^\infty \frac{1}{n}$ be simplified [duplicate]

So I'm doing year 9/10 now and I've just been working with sigmas $\Sigma$. I found across a question which I found quite tricky. Is there a way to write down the answer to this question or make it ...
5
votes
3answers
1k views

Show $\sum_\limits{k=1}^{\infty}1/k$ does not converge. [duplicate]

Show $$\sum_\limits{k=1}^\infty \frac 1 k$$ does not converge. Attempt: Let $s_n=\sum_\limits{k=1}^{n}1/k$, and let $\epsilon=1/2$. For all $N\in\mathbb{N}$, we have $$\left|s_{2n}-s_n\right|=\left|\...
3
votes
2answers
114 views

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 ...
2
votes
5answers
405 views

Let $p_n$ be the sequence defined by $p_n=\sum_{k=1}^n\frac{1}{k}$. Show that $p_n$ diverges even though $\lim_{n\to\infty}(p_n-p_{n-1})=0$ [duplicate]

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}$...
-2
votes
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 ...
3
votes
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 ...
2
votes
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}...
1
vote
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 ...
0
votes
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 ...
1
vote
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.
0
votes
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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