Question

高数计算极限

Answer 1

解：用夹逼定理求解。
∵1≤i≤n时，n^2+6n+5≤n^2+6n+5i≤n^2+6n+5n，∴i/[n^2+6n+5n)≤i/(n^2+6n+5i)≤i/(n^2+6n+5)，对其求和，有lim(n→∞)∑i/[n^2+6n+5n)≤lim(n→∞)∑i/(n^2+6n+5i)≤lim(n→∞)∑i/(n^2+6n+5)，
而∑i=n(n+1)/2，lim(n→∞)∑i/[n^2+6n+5n)=1/2，lim(n∞)∑i/(n^2+6n+5)=1/2，
∴原式=15lim(n→∞)∑i/(n^2+6n+5i)=15/2。
供参考。

Answer 2

解：用夹逼定理求解。∵1≤i≤n时，n^2+6n+5≤n^2+6n+5i≤n^2+6n+5n，∴i/[n^2+6n+5n)≤i/(n^2+6n+5i)≤i/(n^2+6n+5)，对其求和，有lim(n→∞)∑i/[n^2+6n+5n)≤lim(n→∞)∑i/(n^2+6n+5i)≤lim(n→∞)∑i/(n^2+6n+5)，而∑i=n(n+1)/2，lim(n→∞)∑i/[n^2+6n+5n)=1/2