<prompt> Kwa nambari yoyote chanya n, acha langle n rangle iashiria nambari kamili iliyo karibu zaidi na sqrtn. Tathminisum_n 1infty frac2langle n rangle 2-langle n rangle2n.  <response>: Acha n na k ziwe nambari kamili chanya kiasi kwamba langle n rangle k. Kishak - frac12 sqrtn k frac12,auk2 - k frac14 n k2 k frac14.Kwa hivyo, kwa nambari chanya k, thamani za n kiasi kwamba langle n rangle k ni n k2 - k 1, k2 - k 2, dots, k2 k. Kwa hivyo, tunaweza kuandika tena jumla kamaanzaalignsum_n 1infty frac2langle n rangle 2-langle n rangle2n sum_k 1 infty sum_n k2 - k 1k2 k frac2langle n rangle 2-langle n rangle2n sum_k 1infty 2k 2-k kushoto frac12k2 - k 1 frac1 2k2 - k 2 vitone frac12k2 k kulia jumla_k 1infty 2k 2-k cdot frac22k - 1 22k - 2 dots 1 2k2 k sum_k 1infty 2k 2-k cdot frac22k - 12k2 k jumla_k 1infty 2-k2 2k - 2-k2 - 2k 21 - 2-3 20 - 2-8 2-3 - 2-15 2-8 - 2-24 dotsb 3.mwishopanga 