OFFSET
0,2
COMMENTS
Let b(n) = -a(n)/8^n, {b(n)} = {1, -5/4, -3/32, -83/128, -749/2048, -7491/8192, -67415/65536, ...}, then Sum_{n>=0} b(n) is clearly divergent since Sum_{n>=0} a(n)*x^n has radius of convergence 1/16. Let c(n) = A349847(n)/(-4)^n, {c(n)} = {1, -5/2, 11/8, -17/16, 115/128, -203/256, 735/1024, ...}, then Sum_{n>=0} c(n) is the Cauchy product of Sum_{n>=1} b(n) with itself. Since |c(n)| ~ 3/sqrt(Pi*n) and |c(n+1)|/|c(n)| = ((6*n+5)*(2*n-1)) / ((6*n-1)*(2*n+2)) < 1, Sum_{n>=0} c(n) is conditionally convergent by Leibniz's criterion. {b(n)} serves as an example such that the Cauchy product of a divergent series with itself can be conditionally convergent.
LINKS
Wikipedia, Cauchy product
FORMULA
EXAMPLE
Let C(n) denote the Catalan numbers, P = A004981.
a(0) = -P(0) = -1;
a(1) = 2^3 * C(0) * P(0) + P(1) = 10;
a(2) = -2^3 * C(0) * P(1) + 2^5 * C(1) * P(0) - P(2) = 6;
a(3) = 2^3 * C(0) * P(2) - 2^5 * C(1) * P(1) + 2^7 * C(2) * P(0) + P(3) = 332;
a(4) = -2^3 * C(0) * P(3) + 2^5 * C(1) * P(2) - 2^7 * C(2) * P(1) + 2^9 * C(3) * P(0) - P(4) = 1498.
PROG
CROSSREFS
KEYWORD
sign
AUTHOR
Jianing Song, Dec 01 2021
STATUS
approved