Linked Questions

0
votes
7answers
23k 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 ...
10
votes
2answers
955 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
492 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
590 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
473 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 ...
6
votes
3answers
2k 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
139 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
3answers
1k 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 ...
2
votes
5answers
627 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}$...
3
votes
3answers
218 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
640 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
2answers
304 views

Is there an intuitive reason as to why the harmonic series is divergent? [duplicate]

The proof involving partial sums up to the nth term, where n is some power of $2$, completely makes sense. But just looking at the series itself, it seems very strange that it's divergent. For large ...
1
vote
3answers
346 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 the ...
0
votes
2answers
119 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
329 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.

15 30 50 per page