|
|
|
|
|
|
|
|
|
#13 by Michael De Vlieger at Tue Dec 27 16:53:55 EST 2022
|
|
|
|
|
|
|
|
#12 by Peter Luschny at Tue Dec 27 15:22:53 EST 2022
|
|
|
|
|
|
|
|
#11 by Peter Bala at Tue Dec 27 11:07:36 EST 2022
|
|
|
|
|
|
|
|
#10 by Peter Bala at Tue Dec 27 10:39:46 EST 2022
|
|
| LINKS
|
Peter Bala, <a href="/A245637/a245637.pdf">Borel summation of a family of divergent series</a>
|
|
|
|
|
|
|
#9 by Peter Bala at Mon Dec 26 11:01:09 EST 2022
|
|
| FORMULA
|
Equals the Borel sum of the alternating divergent series Sum_{n >= 0} (-1)^n*(2*n + 1)^n. Compare with the alternating convergent series Sum_{n >= 01} (-1)^)^(n/(+1)/(2*n + - 1)^()^n+1) = = Integral_{x = 0..1} x^(x^2) dx. See A253299.
|
|
|
|
|
|
|
#8 by Peter Bala at Mon Dec 26 10:52:48 EST 2022
|
|
| COMMENTS
|
For a, b nonnegative integers, the alternating divergent series Sum_{n >= 0} (-1)^n*(a*n + b)^n is Borel summable to Integral_{x = 1..oo} x^(a-b-1)/x^(x^a) dx.
|
|
|
|
|
|
|
#7 by Peter Bala at Mon Dec 26 10:31:16 EST 2022
|
|
|
|
|
|
|
|
#6 by Peter Bala at Mon Dec 26 10:27:23 EST 2022
|
|
| FORMULA
|
Equals the Borel sum of the alternating divergent series Sum_{n >= 0} (-1)^n*(2*n + 1)^n. Compare with the alternating convergent series Sum_{n >= 10} (-1)^()^n+1)/(/(2*n - + 1)^)^(n = +1) = Integral_{x = 0..1} x^(x^2) dx. See A253299.
|
|
| EXAMPLE
|
0.46230371153732107718203962858827744096102603704840 ......
|
|
| CROSSREFS
|
Cf. A245637, A253299, A359282, A359284, A359285, A359286.
|
|
|
|
|
|
|
#5 by Peter Bala at Sat Dec 24 06:20:02 EST 2022
|
|
| FORMULA
|
Equals the Borel sum of the alternating divergent series 1 - 3^Sum_{n >= 0} (-1 + 5^)^n*(2 - 7^3*n + 9^4 - .... 1)^n. Compare with the alternating convergent series Sum_{n >= 1} (-1 - )^(n+1/3^)/(2 + 1/5^3*n - 1/7^4 + 1/9^5 - ... = )^n = Integral_{x = 0..1} x^(x^2) dx. See A253299.
|
|
| MAPLE
|
evalf(int(1/x^(x^2), x = 01..infinity), 100);
|
|
|
|
|
|
|
#4 by Peter Bala at Sat Dec 24 05:55:25 EST 2022
|
|
| FORMULA
|
Equals the Borel sum of the divergent series 1 - 3^1 + 5^2 - 7^3 + 9^4 - .... Compare with the convergent series 1 - 1/3^2 + 1/5^3 - 1/7^4 + 1/9^5 - ... = Integral_{x = 0..1} x^(x^2) dx . . See A253299.
|
|
|
|
|
|
