第一題好像是這樣
有錯請告知，感謝><
\( \displaystyle \sqrt{1+\frac{1}{1^2}+\frac{1}{2^2}}+\sqrt{1+\frac{1}{2^2}+\frac{1}{3^2}}+\ldots+\sqrt{1+\frac{1}{999^2}+\frac{1}{1000^2}} \)
\( \displaystyle =\sum_{k=1}^{999}\sqrt{1+\frac{1}{k^2}+\frac{1}{(k+1)^2}}=\sum_{k=1}^{999} \sqrt{\frac{k^4+2k^3+3k^2+2k+1}{k^2(k+1)^2}} \)
\( \displaystyle =\sum_{k=1}^{999} \sqrt{\frac{(k^2+k+1)^2}{k^2(k+1)^2}}=\sum_{k=1}^{999}\frac{k^2+k+1}{k(k+1)}=\sum_{k=1}^{999} \left( 1+\frac{1}{k(k+1)} \right)=999 \frac{999}{1000} \)
