<prompt> Kokotoa sum_n1infty kushoto frac2n3n - frac12n kulia.  <response>: Tunaweza kuandika hii upya kama safu mbili tofautisum_n1infty kushoto frac2n3n - frac12n kulia 2sum_n1 infty fracn3n - sum_n1infty frac12n.Ya kwanza, S sum_ n1infty fracn3n frac13 frac29 frac327 dotsb ni mfululizo wa hesabu-jiometri. Kuzidisha kwa 3, kinyume cha uwiano wa kawaida, inatupa3S sum_n1infty fracn3n-1 1 frac23 frac39 frac427 dotsbKuondoa S kutoka 3S inatoabeginaligned 2S 1 frac13 frac19 frac127 dotsb frac11-frac13 frac32.endalignedMfululizo wa pili ni mfululizo wa kijiometri kwa hivyo tunayosum_n1infty frac12n fracfrac121-frac12 1.Kwa hiyo,2sum_n1infty fracn3n - sum_n1infty frac12n frac 32 - 1 frac12. 