若S等于二减二的二次方加二的三次方减二的四次方加二的五次方减二的六次方加...

S=2-2^2+2^3-2^4+2^5-2^6+2^7-2^8+……+2^19-2^20
S=(2-2^2)+(2^3-2^4)+(2^5-2^6)+……+(2^19-2^20)
S=2(1-2)+2^3(1-2)+2^5(1-2)+……+2^19(1-2)
S=-2-2^3-2^5-2^7-2^9-……-2^19---------------(1)
(1)式两边同乘以2^2,得:4S=-2^3-2^5-2^7-2^9-……-2^19-2^21.------(2)
(2)-(1)式,得：3S=-2^21+2,故S=（2-2^21)/3.