<prompt> Kokotoa displaystyle sum_n2infty sum_k1n-1 frack2nk.  <response>: Jumla inachukuliwa juu ya nambari zote chanya n na k kiasi kwamba k le n - 1, au n ge k 1. Kwa hivyo, tunaweza kubadilisha mpangilio wa majumuisho:anzaalignsum_n2infty sum_k1n-1 frack2nk sum_k 1infty jumla_n k 1infty frack2n k sum_k1infty frack2k sum_nk1infty frac12n sum_k 1infty frack2k kushoto frac12k 1 frac12k 2 dotsb kulia sum_k 1infty frack2k cdot frac12k sum_k1infty frack4k.mwishoalignHebuS sum_k 1infty frack4k frac14 frac242 frac3 43 dotsb.Kisha4S 1 frac24 frac342 frac433 dotsb.Tukiondoa milinganyo hii, tunapata3S 1 frac14 frac142 dotsb frac43,hivyo S frac44 9. 