Question

6. Short answer. (a) (2 pts) Find the sum of the series (-1)" =1-1-1-1+1-1+... or explain why it diverges. ( 5) (b) (2 pts) Find the sum of the series or explain why it diverges. n=0

Answer 1

Given that 190 If san is series the we tinal define sequence of partial same {sne such that Si= a 2 die S2 = a ta - af Of Q2 Sn = a + a₂ + agt...fan na1 and en is o Convergent ift sequence of partial soms & S is Convergent. Here he way we can write series as है -LAK+L_ NED not M . . = 52 = S3 - 1 L = 1 - 1 t 0 = 1

and w sur Sy - 11t11 =0 S = soltast Sol dthaltl 1 = or wp-of-t, " Sn = z o I it ir n is even n is odd. dimit of she is exist because it limit o and does not has two . not is Sum of given series exist because on is oort it is a divergent. → because sequence of parlial sums Smais divergente We used here that can is Convergent iff sequence of portial Sum (sno is convergent But s o is divergent so Ecur is also divergent.

③ Given that 120 ton We know that - it - (76) {rn is n=0 geometric Common infinity series where & is ratio the sum of term is given by its at ashere a tor I where a is term. first 1936-16 и со - -------- it is geometric series whose Common ratio is - 5 and first term is then n =o - 16 +5 16 16 so → sum of it is giren series is conrergent.