proposition: Lemma cvg_series_cvg_0 (K : numFieldType) (V : normedModType K) (u_ : V ^nat) : cvgn (series u_) -> u_ @ \oo --> (0 : V).
proof: Proof. move=> cvg_series. rewrite (_ : u_ = fun n => series u_ n.+1 - series u_ n); last first. by rewrite funeqE => i; rewrite seriesSB. rewrite -(subrr (limn (series u_))). by apply: cvgB => //; rewrite ?cvg_shiftS. Qed.
imports: mathcomp(all_ssreflect ssralg ssrint ssrnum matrix) mathcomp(interval rat) mathcomp(mathcomp_extra boolp classical_sets functions) mathcomp(set_interval) reals ereal signed topology normedtype landau