Question

Please help me complete the following proof. Specifically, how does it follow that {s_q} and {t_q} have the same nature given the two cases of convergence and divergence? Thank you!

1. Let (un)nen be a sequence of real numbers and N E N. (a) Prove that ) Un and ) Un have same nature (convergent or divergent). We want to look O nzo n>N at the sequence of partial sums of both series. For q > N, take {sg} as the sequence of partial sums for Un and {tq} as the sequence of partial sums for Un. Thus, n>0 n>N Sq = uo + ... + UN +UN+1 + ... + Uq and tq = un + ... + uq

Answer 1

The nature of a series depends on its sequence of partial sums.

- naN partial Solo Gt (unnen be a sequence of real numbers suppose the two series Eun and Eun ท50 et qUN 6t hall3 = UN+ Untit - - tuq be sequence of 1 sum of Eun and {&q3= Hot 4 t - - t ug be sequence of - sum of ² un N=N parnal nco Sq = 40tu, t --+ Un + Untit -- + uq 4otuit --- + Utuit - 9th term of then sq = 40+ Uit --- + Uwant tą sequence lie. sq3= Uothit + Unes ++43 lie. 6843 = a+6+43 where as yothit ~ + 4N-1 Thus both sequences {sab and {tq3 Conver go and diverges together. and Therefore the series 'un and į un enverges and diverges together net . non