<prompt> Zingatia mlolongo x_nx_n-1-x_n-12, n geq 2, x_1 in0,1. Kokotoa lim _n mshale wa kulia infty fracn2 x_n-nln n .  <response>: Kwa nadharia ya muunganiko wa sauti moja, ni rahisi kuonyesha kuwa x_n mshale wa kulia 0, kama n mshale wa kulia infty. Kisha, kwa nadharia ya Stolz-Cesaro inashikilia beginaligned n x_nfracnfrac1x_n sim frac1 frac1x_n-frac1x_n-1fracx_n-1 x_nx_n-1-x_ n quadfracx_n-1kushotox_n-1-x_n-12kuliax_n-1 21-x_n-1 mshale 1 wa kulia, text as n rightarrow infty . endaligned Badilisha usemi unaozungumziwa na utumie nadharia ya Stolz-Cesaro kwa mara nyingine tena: beginaligned fracn2 x_n-nln n fracn x_n cdotkushoton-frac1x_nkulialn n sim fracn-frac1x_ nln n sim frac1-frac1x_nfrac1x_n-1ln n-ln n-1frac1-frac1x_nfrac1x_n-1-ln kushoto1-frac1 nkulia sim sim nkushoto1-frac1x_n-1-x_n-12frac1 x_n-1kulia-fracn x_n-11-x_n-1 rightarrow-1, text as n rightarrow infty endaligned Jibu: -1 