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极限级数练习题 一、 syms n k x f=(n+1)*(n+2)*(n+3)/(5*n^3) f = ((n + 1)*(n + 2)*(n + 3))/(5*n^3) g=[(-2)^n+3^n]/[(-2)^(n+1)+3^(n+1)] g = ((-2)^n + 3^n)/((-2)^(n + 1) + 3^(n + 1)) h=exp(-x)*atan(x) h = exp(-x)*atan(x) i=(tan(x)-sin(x))/x^3 i = -(sin(x) - tan(x))/x^3 f1=limit(f,inf) f1 = 1/5 g1=limit(g,inf) g1 = 1/3 h1=limit(h,inf) h1 = 0 i1=limit(i,0) i1 = 1/2 二、 syms n f=1/(n*(n+1)) f = 1/(n*(n + 1)) g=sqrt(n/(n+1)) g = (n/(n + 1))^(1/2) h=sin(pi/n*(n+1)) h = sin((pi*(n + 1))/n) i=n*3^n/2^n i = 1/2^n*3^n*n f1=symsum(f,n,1,inf) f1 = 1 g1=symsum(g,n,1,inf) g1 = sum((n/(n + 1))^(1/2), n == 1..Inf) h1=symsum(h,n,1,inf) h1 = (sum(exp(-pi*i)*exp(-(pi*i)/n), n == 1..Inf)*i)/2 - (sum(exp(pi*i)*exp((pi*i)/n), n == 1..Inf)*i)/2 i1=symsum(i,n,1,inf) i1 = Inf 因而，第（1）小题收敛，（2）（3）（4）小题均发散. 三、 syms n s=(-1)^n*(n+1)/3^n s = (-1)^n/3^n*(n + 1) f=(-1)^n/log(n) f = (-1)^n/log(n) h=n/2^n*cos(2*n*pi/3) h = 1/2^n*n*cos((2*pi*n)/3) s1=symsum(s,n,1,inf) s1 = -7/16 f1=symsum(f,n,2,inf) f1 = sum((-1)^n/log(n), n == 2..Inf) h1=symsum(h,n,1,inf) h1 = ((3^(1/2)*i)/2 - 1/2)/(4*(3^(1/2)*(i/4) - 5/4)^2) - ((3^(1/2)*i)/2 + 1/2)/(4*(3^(1/2)*(i/4) + 5/4)^2) 因而，第（1）、（3）小题收敛，第（2）小题不收敛. s2=symsum(abs(s),n,1,inf) s2 = 5/4 f2=symsum(abs(f),n,2,inf) f2 = sum(abs((-1)^n)/log(n), n == 2..Inf) h2=symsum(abs(h),n,1,inf) h2 = sum(1/2^n*n*abs(cos((2*pi*n)/3)), n == 1..Inf) 因而，第（1）小题绝对收敛，第（2）小题不收敛，第（3）小题条件收敛. 四、 syms n x s=((2*n-1)*(x^(2*n-2)))/(2^n) s = 1/2^n*x^(2*n - 2)*(2*n - 1) f=(x^n)/(n*2^n) f = (1/2^n*x^n)/n g=n*(n+1)*x^(n-1)/2 g = (n*x^(n - 1)*(n + 1))/2 h=x^n/(n*(n+1)) h = x^n/(n*(n + 1)) s1=symsum(s,n,1,inf) s1= 1/4*pi^(1/2)*2^(3/4)*(x^2)^(1/4)/(1-1/2*x^2)^(3/2)*LegendreP(1/2,1/2,(1/2*x^2+1)/(1-1/2*x^2)) f1=symsum(f,n,1,inf) f1= -log(1-1/2*x) g1=symsum(g,n,1,inf) g1= -1/(x-1)^3 h1=symsum(h,n,1,inf) h1= 1-(x-1)/x*log(1-x) 五、 1、 syms n f=((3^n+5^n)/n)^(1/n) f = ((3^n + 5^n)/n)^(1/n) s=limit(f,n,inf) s = 5 R=1/5 R = 0.2000