<prompt> Acha F0 0, F1 frac32, naFn frac52 Fn - 1 - Fn - 2kwa n ge 2. Tafutasum_n 0infty frac1F2n.  <response>: Tunadai kuwa Fn 2n - frac12n kwa nambari zote zisizo hasi n. Tunathibitisha hili kwa uingizaji mkali.Matokeo ya n 0 na n 1. Chukulia kuwa matokeo yanashikilia n 0, 1, 2, dots, k, kwa nambari fulani isiyo hasi k ge 1, hivyo Fk - 1 2k - 1 - frac12k - 1 na Fk 2k - frac11 2k.KishaanzaalignFk 1 frac52 Fk - Fk - 1 frac52 kushoto 2k - frac12k kulia - kushoto 2k - 1 - frac12 k - 1 kulia frac52 cdot 2k - frac52 cdot frac12k - frac12 cdot 2k frac22k 2 cdot 2k - frac12 cdot frac12k 2k 1 - frac12k 1.endalignKwa hivyo, matokeo hushikilia n k 1, kwa hivyo kwa introduktionsutbildning, matokeo hushikilia n zote ge 0.Kisha jumla tunayotafuta nisum_n 0infty frac1F2n sum_n 0infty frac122n - frac122n sum_n 0infty frac22n22n2 - 1.Acha x 22n. Kishaanzaalignfrac22n22n2 - 1 fracxx2 - 1 fracx 1 - 1x2 - 1 fracx 1x2 - 1 - frac1x2 - 1 frac1x - 1 - frac1x2 - 1 frac122n - 1 - frac122n 1 - 1.endalignKwa hivyo, jumla ya darubini zetu:anzaalignjumla_n 0infty frac22n22n2 - 1 sum_n 0infty kushoto frac122n - 1 - frac122n 1 - 1 kulia kushoto frac1220 - 1 - frac1221 - 1 kulia kushoto frac1 221 - 1 - frac1222 - 1 kulia kushoto frac1222 - 1 - frac1223 - 1 kulia dotsb 1.mwishopanga 