Question

3. Suppose lim s(a) dr = co, where f(a) is a positive, decreasing and continuous function. Which of the following statements is true about the series f(n)? Choose one. n=1 *Please write the letter of your choice. (a) The series converges too. (b) The series converges, but not necessarily to o. (c) The series diverges. (d) The given information is not enough to determine if the series converges or diverges.

Answer 1

Lim food - OG Si fos - Consider Q (B) - J B fins da is unbounded. for B₂ > B, 7 (B2) — 9(6.) B2 fronda zo B. [since few yo \ xx4] - monotone is o n Let Sn = KE increasing in [!") #kein since f(K)70 fo) {sn) is monstore of parttal increasing Sume and the infinite series I find is divergent if Ssa? is unbounded above. Sequence od n여 natural number a Let Byt .. Then there exist n such that n< Bntl.

As As from yo for 26 (1) J for de s S for de s B ntl fra dx 1 fore ac [k, k+] (kent). Since f is morotone decreasing f(K+) s from s fex) J fra ty da es K+1 kt! K+1 =) fondas Pek) da k k к KTI F (k+i) s fox da s f(x) - T k n from Ź f(x+1) $ kt! El fruida a Ŝ fra KT ka May ka! nel = Snei - foins frojda En ntl =) Sn7 fonda » Q (13) (from

Since is unbounded above t By, 1 (3) 9 (6) then from Hence {fen) {sn) is unbounded above is divergent nal option © is correct