3
votes
2answers
95 views

Sum of divergent series

I saw a lot of article in Math SE like Why does 1+2+3+⋯=−1/12? and S=1+10+100+100+10000+…=−1/9? How is that and lot of others. Also I saw this one of Ramanujan summation but I do not get the ...
-1
votes
1answer
35 views

Is there an expression for $(S/k)$ where $S=\sum_{n=1}^\infty n$ and $k \in \mathbb{Z}$?

Given that $S=\sum_{n=1}^\infty n=-1/12$ (for an explanation see this question or this video from Youtube) For example if $k=4$: $(S/4)=1/4+2/4+3/4+1+5/4+6/4+7/4+2+9/4...$ Please edit to improve ...
1
vote
2answers
184 views

Is $1 + 2 + 3 + \dots = -\frac{1}{12}$ really true? [duplicate]

I've read this strange result of the sum of all positive integers being $-\frac{1}{12}$. Is it really true? Does this also mean this is true? $$\sum_{n=1}^k n = \frac{k\cdot(k+1)}{2}$$ ...
0
votes
0answers
45 views

Dirichlet series summation method

Is it true that if $\lim\limits_{x\to1^-}a_1x+a_2x^2+a_3x^3+...=C$ exists then it is necessarily true that $\lim\limits_{s\to0}\frac{a_1}{1^s}+\frac{a_2}{2^s}+\frac{a_3}{3^s}+... =C$ It seems like ...
7
votes
0answers
131 views

A Ramanujan-like summation: is it correct? Is it extensible?

I'm still exercising with summation-procedures which I try to make correct Ramanujan-summations. Looking at the series $$ s(1/2,2) = (1/2)+(1/2)^4+(1/2)^9+(1/2)^{16}+... $$ and more general $$ s(b,p) ...
0
votes
2answers
95 views

Does sum of all natural numbers contradict another rule?

I must say that I am not a mathematician, just a enthusiast who likes to read all the "weird" results in mathematics. I read that sum of all natural number equals to $-1/12$ and I am also aware that ...
2
votes
2answers
201 views

What consistent rules can we use to compute sums like 1 + 2 + 3 + …?

$ \newcommand{ifelse}[3]{ \left( \begin{cases} #1\text{ if }#2 \\ #3\text{ otherwise} \end{cases} \right) } $ A recent Numberphile video on 1+2+3+... has made this question ("Why?") popular again, as ...
3
votes
4answers
231 views

$2 - 4 + 6 - 8 + \cdots = 1 - 1 + 1 -1 + \cdots$?

I want to talk about the weirdness of $2\sum\limits_{n=1}^\infty n(-1)^{n-1}$ , $$\sum\limits_{n=1}^\infty n(-1)^{n-1} = 1 - 2 + 3 - 4 + 5 - 6 + \dots$$ $$\times 2 \implies 2\sum\limits_{n=1}^\infty ...
6
votes
2answers
130 views

What can be computed by axiomatic summation?

Here are three simple properties one might require of a summation method for divergent series: A stable summation scheme is one in which (assuming also each sums are defined iff the other is) ...
4
votes
2answers
164 views

What are some physical, geometric, or otherwise useful interpretations of divergent sums?

Is there any practical application to discussing the 'sum' of sequences that are not convergent under Cesàro summation? Why would we want to assign a value to an otherwise divergent sequence and ...
3
votes
0answers
104 views

Prove the divergence of the sum of the reciprocals of practical numbers

A practical number is an integer $n$ for which every smaller integer can be expressed as a sum of distinct divisors of $n$. How can we prove that the sum of their reciprocals is divergent? Are there ...
5
votes
4answers
153 views

Summation of series $\sum_{n=1}^\infty \frac{n^a}{b^n}$?

How can we evaluate this series $$\sum_{n=1}^\infty \frac{n^a}{b^n}?$$ Here $a$ and $b$ are positive integers. If $b=1$ then series will be diverging, in other cases, it will be converging, but how ...
7
votes
4answers
257 views

Sum of kth roots ($\sum\sqrt[k]{m}$)

I'm trying to find an asymptotic to $$S(n) = \sum_{k=1}^n\sqrt[k]{m}$$ From computational tests, it seems to grow nearly as slowly as $n$. However even $$\sum_{k=1}^\infty\sqrt[k]{m}-1$$ diverges (for ...
2
votes
0answers
110 views

Proof for a summation-procedure using the matrix of Eulerian numbers?

I've discussed a procedure for divergent summation using the matrix of Eulerian numbers occasionally in the last years (initially here, and here in MSE and MO but not in that generality and thus(?) ...
17
votes
1answer
727 views

Prove that sum is finite

Let $j \in \mathbb{N}$. Set $$ a_j^{(1)}=a_j:=\sum_{i=0}^j\frac{(-1)^{j-i}}{i!6^i(2(j-i)+1)!} $$ and $a_j^{(l+1)}=\sum_{i=0}^ja_ia_{j-i}^{(l)}$. Please help me to prove that the following sum is ...