Questions tagged [divergent-series]
128 questions
10
votes
1 answer
ζ(-n) and "powers" of Grandi's series
For n a non-negative integer, $ζ(-n)$ can be interpreted as assigning a value to the (divergent) series $1^n+2^n+3^n+4^n+\cdots$
A value can also be assigned to the related series ${n+0 \choose n}+{n+1 \choose n}+{n+2 \choose n}+{n+3 \choose…
Robin Saunders
- 3,512
8
votes
4 answers
Summing a divergent series and a constant combined
At least according to the answer to this question, $\zeta(1) = \gamma $ (once reqularized, of course).
Let me rephrase that by stating that:
$$ \sigma(\zeta(1)) = \gamma $$
Here, $\sigma(x)$ is the 'summation-function'. It's a function that assigns…
Max Muller
- 4,505
6
votes
3 answers
Summating divergent series arising from the application of the Euler-Maclaurin formula to power law functions with non-integer exponents
As the title says, I'm stuck trying to find an expression for $\sum_{n=a}^b n^{q}$, with q being a positive rational number but not an integer, which does not demand unfeasibly long computation times for large ranges of $b-a$. From the application…
MarioVX
- 61
3
votes
2 answers
A family of divergent series
Is it reasonable to assert that $0^k - 1^k - 2^k + 3^k - 4^k + 5^k + 6^k - 7^k - ... = 0$ for all $k > 1$?
Here the signs are given by the Thue-Morse sequence; that is, the sign of $m^k$ is $+$ or $-$ according to whether the number of 1's in the…
James Propp
- 19,363
1
vote
1 answer
Partitions of $\mathbb N$ which generate only divergent series of reciprocals of elements of those sets
This question is a result of some thinking about $\mathbb N$, divergent series and partitions of sets.
Although elementary, I am not skilled enough to answer it at the present moment.
Is it possible to partition $\mathbb N$ into an infinite number…
user147968
0
votes
1 answer
Why is it customary to have formal series at infinity in the context of resurgence and 1-summability?
In the context of the theory of 1-summability and resurgence, it is customary to deal with formal series "at infinity" rather than at $0$
$$
\sum_{n=0}^{\infty}a_nz^{-n}
$$
This is stated for example at the beginning of section 3 on…
PhoenixPerson
- 247