Question

请问这两个极限怎么求....

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Answer 1

　　解：第1题，lim(n→∞)(2^n+3^n)^(1/n)=e^[lim(n→∞)(1/n)ln(2^n+3^n)]，
　　而lim(n→∞)(1/n)ln(2^n+3^n)=lim(n→∞)[(ln2)2^n+(ln3)3^n]/(2^n+3^n)=lim(n→∞)[(ln2)(2/3)^n+ln3]/[1+(2/3)^n]=ln3，∴lim(n→∞)(2^n+3^n)^(1/n)=3。
　　第2题，用夹逼定理求解。∵当1≤k≤n时，n^2+n+1≤n^2+n+k≤n^2+n+n，∴k/(n^2+n+n)≤k/(n^2+n+k)≤k/(n^2+n+1)，k=1,2，……，n。∴∑k/(n^2+n+n)≤∑k/(n^2+n+k)≤∑k/(n^2+n+1)。
　　而lim(n→∞)∑k/(n^2+n+n)=(1/2)lim(n→∞)n(n+1)/(n^2+n+n)=1/2，lim(n→∞)∑k/(n^2+n+1)=(1/2)lim(n→∞)n(n+1)/(n^2+n+1)=1/2，∴lim(n→∞)∑k/(n^2+n+k)=1/2。
　　供参考。