考研真题十二?1.设级数收敛,则必收敛的级数为( ).n?un00数一考研题?1?(A)?(?1)nun;(B)n?1nn??u2?1n;??(C)u2n?1?u2n);(D)un?(?1n?(?1n?un?1).?2.求幂级数n?1xn?13n?(?2)nn的收敛区间,并讨论该区间端点处的收敛性.01数一考研题3.?设f(x)?{1xx2arctanx,x?0,试将f(x)展开成x的幂级数,并求1,x?0n??(?1)n级数的和.00数一考研题?11?4n24.设u2,3,?),且nn?0(n?1,nlim??u?1,则级数02数一考研题n?(?1)n?1n??1(1u?1nun?1)(A)发散;(B)绝对收敛;(C)条件收敛;(D)不能判定.?5.设x2??),则a2?______.03数一考研题n?ancosnx(???x??0??(?1)n6.将函数f(x)?arctan1?2x1?2x展开成x的幂极数,并求级数n?02n?1的和.03数一考研题n7.设a3n?1n?2?10xn1?xndx,则极限nlim??nan等于()03数二考研题33(A)(1?e)2?1;(B)(1?e?1)2?1;33(C)(1?e?1)2?1;(D)(1?e)2?1.?8.设?an为正项级数.下列结论中正确的是( ).04数一考研题n?1.41.(A)若minnan?0,则级数n??n??an收敛;?1?(B)若存在非零常数?,使得nlim??nan??,则级数an发散;n??1?(C)若级数?an收敛,则limn2a??n?0;n?1n?(D)若级数n?an发散,则存在非零常数?,使得limnan??.?1n??9.设有方程xn?nx?1?0,其中n为正整数.证明方程存在惟一实根xn,?并证明当??1时,级数收敛.04数一考研题n?xn??110.??求幂级数(?1)n?1???1?1?2nn?1?n(2n?1)???x的收敛区间与和函数f(x).05数一考研题11.?若级数?an收敛,则级数( )06数一考研题n?1(A)??a(B)??(?1)nan?1n收敛;n收敛;n?1(C)??a?anan?1收敛;(D)n?an?1n?1?n?12收敛.12.将f(x)?x2?x?x2展成为x的幂级数.06数一考研题13.设幂级数n??a?0nxn在(??,??)内收敛,其和函数y(x)满足07数一考研题y???2xy??4y?0,y(0)?0,y?(0)?1.(Ⅰ)证明an?2?2n?1an,n?1,2,?;(Ⅱ)求y(x)的表达式.14.设函数f(x)在(0,??)上具有二阶导数,且f??(x)?0,令un?f(n)(n?1,2,?),则下列结论正确的是( ).07数一、数二考研题(A)若u1?u2,则{un}必收敛; (B)若u1?u2,则{un}必发散;(C)若u1?u2,则{un}必收敛; (D)若u1?u2,则{un}必发散..42.15.已知幂级数n?0?a?n(x?2)n在x?0处收敛,在x??4处发散,则幂级数n?0?a?n(x?3)n的收敛域为__________.08数一考研题16.将函数f(x)?1?x2(0?x??)展开成余弦形式的傅里叶级数,并求n?1??(?1)n2n?1的和.n??08数一考研题17.设有两个数列{an},{bn},若liman?0,则( ).(A)当(B)当(C)当(D)当09数一考研题?b?bn?1?n?1??n收敛时,发散时,n?1?nn?1?a?a?n?1??nbn收敛;发散;收敛;发散.nbn??n?1n?1?bn收敛时,bn发散时,?a?aS1?22nbn22nbnn?118.设an为曲线y?xn与y?xn?1(n?1,2,?)所围成区域的面积,记?a,S??an2n?1n?1??2n?1,09数一考研题求S1与S2的值.(?1)n?12n19.求幂级数?x的收敛域及和函数.n?12n?1?10数一考研题.43.