Question

Let 4. ) Using only the definition of infinite Series convergence, prove the following: w, ZER. Given of in and on respectively convergent to X and Y, then zyn =wX t zY In are 2 WXN t DE 6 Use the theorem above to prove the following: Let WEIR. Given to and are respectively convergent to X and Y, then £ w x n = wX,

Answer 1

Play Definition : Xet be an ENT infinite series Let us define Si=w S₂ = Uituz -- Su= lituzt-. tun nel be ha {suca is Then their sequence {sus is said to be the sequence of partial sums of the series un The infinite series un is said to as the sequence convergent or divergent according converge divergent. The series {un converges to u iff lim Sn=u. It is given that an and Zon are converge y respectively Let us to consider the sequences of partial sums {snsman and string off and 2 on respectively. Sn = datat-tkn , ny, and tn= Yixdat ---tyn X and lim tn Y n=1 m) to X and , nyl Then lim Sn no Now consider the sequence of partial. Sums of the series ž (want zyn : n=1 Pn = (w 24 + Z81)+(Wx2+ Z42) +-not (94+22+-- + r) + 2 (fit7z+--+*n) (went zyn) w sytetu

P Where w and 7 are Now & {Pusoma is defined as Pn= W Snt ztn inyl fixed real numbers. lim Pn = lim (Wsnt Ztn) w lim Sut z lim tn = wx+zy (by (*)) Therefore the sequence of partial sums of the series want zyn converges to wx+Z Y -100 n+$ n>DO n>DO N=1 Da W=1 CO ire. Na It is nal na Hence Ž want zyn converges to wX+ZY Swrentz wX+zy (proved) given that the series 8xn and a yn converge to X and y respectively, OER Therefore by the result (a), we get wanto. In wX+0.7 WE TR Also n=1 sie. wan = WX (proved) na