Recently, sources for mathematical infotainment, for example , have given some information on the interpretation of divergent series as real numbers, for example  $\sum_{i=0}^\infty i = -{1 \over 12}$  This equation in particular is said to have some importance in modern physics, but, being infotainment, there is not much detail beyond that.  As an IT Major, I am intrigued by the implications of this, and also the mathematical backgrounds, especially since this equality is also used in the expansion of the domain of the Riemann-Zeta function.  But how does this work? Where does this equation come from, and how can we think of it in more intuitive terms?
1958
$\displaystyle\sum_{n=1}^\infty \frac{1}{n^s}$ only converges to $\zeta(s)$ if $\text{Re}(s) &gt; 1$.  Why should analytically continuing to $\zeta(-1)$ give the right answer?
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$$\sum_2^\infty e^{3-2n}$$  The formulas for these things are so ambiguous I really have no clue on how to use them.  $$\frac {cr^M}{1-r}$$  $$\frac {1e^2}{1-\frac{1}{e}}$$  Is that a wrong application of the formula and why?
113204
How could the series below be computed ? $$\sum_{k=0}^{\infty}\frac{1}{2^{k!}}$$ It's not a series from a book, but a series I thought of many times, and I didn't manage to figure out what I should do here. I'm just curious to know if there are some known ways for approaching such a series. Thanks!
52305
I was doing some problems in algebraic number theory and this series came up  $$\sum_{n=0}^\infty\frac{1}{9n^2+9n+2}.$$  So, I would like to know the value of this series. However, I don't want a full answer, only hints. I have tried to use Fourier's series, but with no success, perhaps an answer can be gotten usign this.  Thanks
116233
I was going through some old stuff when I happened upon fractional summations.  They come of the form $\sum_{k=a}^bf(k)$ for $b-a\notin\mathbb Z$.  One such example is the extended harmonic numbers, which has: $$\sum_{k=1}^n\frac1k=\sum_{k=1}^\infty\frac1k-\frac1{k+n}$$ Which got me thinking.  If we had $f(x)$ be continuous and $\lim\limits_{x\to\infty}f(x)=0$, then one might perchance define fractional summations as follows: $$\sum_{k=1}^xf(k)\equiv\sum_{k=1}^\infty f(k)-f(k+x)\forall x\notin\mathbb Z$$ For example, $$\sum_{k=1}^x\frac1{k^s}=\sum_{k=1}^\infty\frac1{k^s}-\frac1{(k+x)^s}=\zeta(s)-\zeta(s,x+1),s\ne1,s&gt;0$$  This got me thinking:  What other ways can we extend summations to fractional orders? Anything to worry about the above definition? Where would these fractional summations appear?   Related:
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I came across following  bi-infinite sum:  $\sum_{k=-\infty}^{\infty} 2^k x^{2^k}$  Is this a known series? After some plotting I have the feeling that it could be equal or very similar to  $-\frac{1}{\ln(2) \ln(x)}$  for all $x\in(0,1)$. Is my assumption correct?
3701
I was wondering if the following is true. Suppose a series $a_n$ is greater than 0 for all positive integer n, and that $\sum \frac {a_n}n$ converges, then is $\displaystyle \lim_{m\to \infty}\sum_{n= 1}^m {a_n \over m+n} = 0$? It seems to be true because if $\sum {a_n\over n}$ converges, then that means that ${a_n\over n }\to 0$ for $n\to \infty$. This means, neglecting $n$, ${a_n \over m+n}$ will also tend to 0, and thus the summation would be equal to 0, but I don't know if this is true.
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I am a Biology major (please don't shame me!) but I really enjoy mathematics. Recently I have been reading about this conjecture and its importance in understanding the distribution of primes. After being studied for so long, how come all attempts have failed? Why is this such a difficult thing to study and work on? Would finding a closed form of $$\sum_1^\infty \frac {1}{n^s}$$ help the issue? Have any attempts been made at this at all? Thank you very much for your time educating me and your help!!
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As the title says, given a series $b_n &gt; 0$, where $\sum_{n=1}^\infty b_n$ is divergent:  Show that the series $$\sum_{n=1}^\infty \frac{b_n}{1+b_n}$$ is also divergent.  So I've defined the series as $\sum_{n=1}^\infty u_n v_n$ where $u_n = b_n$ and $v_n = \frac {1}{1+b_n}$.   I'm sure there was some sort of convergence test which stipulated that for a product of two series, if one diverges the entire thing does, however I might be mistaken. If so, how do I properly prove this?
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I'm trying to show that the series $\sum_{n=1}^\infty\frac{n}{n^2+1}$ is divergent.   I'm trying to use a comparison test to do so. To do so, I first can see that $\sum_{n=1}^\infty\frac{n}{n^2+1} \geq \sum_{n=1}^\infty\frac{n}{n^2+n} = \sum_{n=1}^\infty\frac{1}{n+1}$ And I can use a change of variable $k = n+1$, so show $\sum_{n=1}^\infty\frac{1}{n+1} = \sum_{k=2}^\infty\frac{1}{k} = \infty$. Then since $\sum_{n=1}^\infty\frac{n}{n^2+1} \geq \sum_{k=2}^\infty\frac{1}{k}$ it must be that $\sum_{k=2}^\infty\frac{1}{k}$ also diverges to infinity. Is this a valid way to show it?
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$$\sum_{n=1}^\infty \frac 1{n!}$$   I think we may compare it with $\frac{1}{n^n}$, and say that $\frac{1}{n!}&gt;\frac{1}{n^n}$. But I am not sure of the convergence or divergence of $\frac{1}{n^n}$ itself, so this turns out to be useless.  What other way is there?
106805
So I just started learning series, and I am having trouble understanding them. I have a series below:  $$\sum_{n=0}^{\infty} \frac{1}{(2n+1)2^n}$$  Can someone explain to me how I should go about this, from start to finish? I couldn't find any online websites that explained how to find sums, so if you have that, it would be greatly appreciated!
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For example how come $\zeta(2)=\sum_{n=1}^{\infty}n^{-2}=\frac{\pi^2}{6}$. It seems counter intuitive that you can add numbers in $\mathbb{Q}$ and get an irrational number.
81871
I recall from complex analysis a while ago that the series $$\sum_{n=1}^\infty \frac{z^{n!}}{n}$$ has radius of convergence 1, diverges on roots of unity, and converges elsewhere on the circle. It's simple enough to see that it has to diverge on roots of unity: consider $z$ such that $z^k = 1$ for some $k \in \mathbb{N}.$ Plug this into the series and you see that past the $k$'th term, $z^{n!} = (z^k)^m = 1$ for some $m \in \mathbb{N}$; hence, the series becomes harmonic and must diverge.  But I don't recall how to show that the series converges for other values on the circle. If this is simple I'd like just a hint, if it is involved I would prefer a sparse outline of the proof. Thank you!
67159
Suppose $\sum_{n=1}^\infty \frac{1}{a_n} = A$ is summable, with $a_n &gt; 0,$ $n = 1,2,3,\cdots.$ How can we prove that $\sum_{n=1}^\infty \frac{n}{a_1 + \dots + a_n}$ is also summable?  This question came from a problem-solving seminar, but I'm quite stuck without a push in the right direction. I tried a few things, including Cauchy-Schwarz (which says $\sum_{n=1}^\infty \frac{n}{a_1 + \dots + a_n} &lt; \sum_{n=1}^\infty \frac{A}{n}$) and also the idea of assuming the latter series diverges and attempting to deduce the divergence of the former series from that, using facts such as $\sum a_n = \infty \implies \sum \frac{a_n}{a_1 + \cdots + a_n} = \infty$. Nothing has worked so far.
106849
(I was requested to edit the question to explain why it is different that a proposed duplicate question. This seems counterproductive to do here, inside the question it self, but that is what I have been asked by the site and moderators. There is no way for me to vote against their votes. So, here I go: Please stop voting this as a duplicate so quickly, which will eventually lead to this question being closed off. Yes, the other question linked to asks the same math, but any newcomer to the problem who was exposed to it via physics, as I was, will prefer this question instead of the one that is purely mathematically. I beg the moderators to not be pedantic on this one. This question spills into physics, which is why I did the cross post to the physics forum as well.)  How does the sum of the series “1 + 2 + 3 + 4 + 5 + 6…” to infinity = “-1/12”, in the context of physics?  I heard Lawrence Krauss say this once during a debate with Hamza Tzortzis (). I found a transcript of another debate between Krauss and William Lane Craig which has the same sum. Here is the paragraph in full:     Let’s go to some of the things Dr. Craig talked about. In fact, the   existence of infinity, which he talked about which is   self-contradictory, is not self-contradictory at all. Mathematicians   know precisely how to deal with infinity; so do physicists. We rely on   infinities. In fact, there’s a field of mathematics called “Complex   Variables” which is the basis of much of modern physics, from   electro-magnetism to quantum mechanics and beyond, where in fact we   learn to deal with infinity; without the infinities we couldn’t do the   physics. We know how to sum infinite series because we can do complex   analysis. Mathematicians have taught us how. It’s strange and very   unappetizing, and in fact you can sum things that look ridiculous. For   example, if you sum the series, “1 + 2 + 3 + 4 + 5 + 6…” to infinity,   what’s the answer? “-1/12.” You don’t like it? Too bad! The   mathematics is consistent if we assign that. The world is the way it   is whether we like it or not.   -- Lawrence Krauss, debating William Lane Craig, March 30, 2011  Source:   CROSS POST: I'm not sure if I should post this in mathematics or physics, so I posted it in both. Cross post:   EDIT: I did not mean to begin a debate on why Krauss said this. I only wished to understand this interesting math. He was likely trying to showcase Craig's lack of understanding of mathematics or logic or physics or something. Whatever his purpose can be determined from the context of the full script that I linked to above. Anyone who is interested, please do. Please do not judge him out of context. Since I have watched one of these debates, I understand the context and do not hold the lack of a full breakdown as being ignorant. Keep in mind the debate I heard this in was different from the debate above.
15494
Determine whether the series $\sum_{n=1}^{\infty}(\frac{n}{n+1})^{n^2}$ is convergent or divergent.  I've tried applying all the basic tests to no avail. I need to find out what the "trick" for this particular series is.