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**alexandrabel90** given that the seq {an} is non increasing and an tends to 0, i need to show that sum from n=1 to infinity of (-1)^n an conversed. this is the alternating series theorem..

i was reading its proof and wondering,

if we consider only the even terms of the seq,

then S2n= -a1 + (a2 -a3) +(a4... = by paring, all pairs of terms are positive except -a1

then does it mean that S2n is increasing and hence bounded below by -a1?

then we cant say that the partial sum converges right?

may i know what is wrong with my understanding?

(my notes state that by the monotone convergence thm, S2n converges to a limit as n tend to infinity)