Question

如何判断敛散性？

Answer 1

极限审敛法:\x0d\x0a∵lim(n→∞)n*un=(3/2)^n=+∞\x0d\x0a∴un发散.\x0d\x0a\x0d\x0a比值审敛法:\x0d\x0aun+1=3^(n+1)/[(n+1)*2^(n+1)]=3^n*3/[(n+1)*2^n*2]\x0d\x0aun+1/un=3n/(2n+2)\x0d\x0alim(n→∞)un+1/un=3/2>1,∴发散\x0d\x0a\x0d\x0a根值审敛法:\x0d\x0an^√un=3/2*n^√(1/n)=3/2*(1/n)^(1/n)\x0d\x0a令t=1/n,则当n→∞时t→0,t^t→1\x0d\x0a∴lim(n→∞)n^√un=3/2>1,发散.