0
votes
0answers
26 views

Find the closed-form of a series

Suppose that $x$ is positive number such that $x>0$. I just wonder is there existing a closed form of the series below $f(x)=\sum_{l=0}^{\infty}(2l+1)e^{-xl(l+1)}$. Is the well-known ...
0
votes
1answer
61 views

Concerning the sum $\sum_{n = 1}^\infty \sin nx$

I recently came across this question and I posted an answer. It has been pointed out that my answer is incorrect. I cannot work out what is wrong with my reasoning. The answer I gave corresponds with ...
1
vote
1answer
44 views

Prove Convergence or Divergence

I just need to prove either convergence or divergence for this. Having some serious trouble and would appreciate all help! $$\sum_{n=1}^{\infty}\frac1{n^{1/3}(1+n^{1/2})}$$
4
votes
2answers
178 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
37 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
210 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? $$\sum_{n=1}^k n = \frac{k\cdot(k+1)}{2}$$ ...
1
vote
0answers
71 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 ...
8
votes
0answers
195 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
110 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
233 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
237 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
140 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
198 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
113 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
166 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
265 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
116 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
730 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 ...