｢a·x˄r Σr=0,∞｣
= a + a·x + a·x² + ... (a≠0)
converges if and only if |x|<1.
Moreover, its sum is then a/(1−r).ᐥ
Partial sum,
(1−x)·｢a·x˄r Σr=0,n−1｣
= ｢a·x˄r Σr=0,n−1｣
− ｢a·x˄(r+1) Σr=0,n−1｣
= ｢a·x˄r Σr=0,n−1｣
┆｢a·x˄(r+1) Σr=0,n−1｣=｢a·x˄r Σr=1,n｣┆
− ｢a·x˄r Σr=1,n｣
= a+｢a·x˄r Σr=1,n−1｣
− (｢a·x˄r Σr=1,n−1｣+a·x˄n)
= a−a·x˄n
For r≠1,
｢a·x˄r Σr=0,n−1｣ = (a−x˄n)/(1−r)
For |r|<1,
｢a·x˄r Σr=0,∞｣
= ｢｢a·x˄r Σr=0,n−1｣ Ƚn￫∞｣
= ｢(a−x˄n)/(1−r) Ƚn￫∞｣
┆｢x˄n｣ is a null sequence.┆
= a/(1−r)
