<prompt> Tafuta nambari kamili iliyo karibu zaidi na 1000sum_n310000frac1n2-4.  <response>: Kumbuka hilofrac1n2-4 frac1n-2n2 frac14leftfrac11 n-2 - frac1n2kulia.Kwa hivyo, jumla ya darubini zilizotolewa: anzaalign 1000sum_n310000frac1 n2-4 1000 cdot frac14 sum_n310000 kushotofrac1n-2 - frac1 n2kulia 250 kushotofrac11 frac12 frac13 frac14 - frac19999 - frac110000 - frac110001 - frac110002kulia 250 125 83. muhtasari3 62.5 - varepsilon endalignedambapo varepsilon 250lefttfrac19999tfrac110000tfrac110001 tfrac110002kulia. Hii hurahisisha hadi 520.8overline3 - varepsilon, na kwa hivyo jibu ni 521.Ili kuangalia kuwa varepsilon ni ndogo vya kutosha isiathiri jibu, tunaweza kuandika varepsilon 250 cdot 4 cdot frac15000 0.2. Hii inaonyesha kuwa jumla ni kati ya 520.8 overline3 na 520.6jumla3, na kwa hivyo nambari kamili iliyo karibu ni 521, kama ilivyoelezwa hapo awali. 