<prompt> Compute lim _n mshale wa kulia infty sum_k1nkushotofrackn2kuliafrack n21 .  <response>: Tunatumia ukweli kwamba lim _x rightarrow 0 xx1 . Kwa sababu hiyo, tunayo lim _x rightarrow 0 fracxx1x1 . Kwa tatizo letu, acha epsilon0 iwe nambari ndogo chanya isiyobadilika. Kuna nepsilon kiasi kwamba kwa nambari yoyote n geq nepsilon, 1-epsilonfracleftfrackn2 kuliafrackn21frackn21epsilon, quad k1,2, nukta, n. Kutokana na hili, kwa kutumia sifa za uwiano, tunapata 1-epsilonfracsum_k1nleftfrackn2kulia frackn21sum_k1n frackn21epsilon, quad text for n geq nepsilon . Kwa kujua kwamba sum_k1n kfracnn12, hii ina maana 1-epsilon fracn1 2 nsum_k1nkushotofrackn2kuliafrackn21 1epsilon fracn12 n, quad text for n geq nepsilon . Inafuata kwamba lim _n mshale wa kulia infty sum_k-1nleftfrackn2kuliafrac kn21frac12 . 