Example 8 Let $a_{n}=\frac{1}{(n+1) \sqrt{n}+n \sqrt{n+1}}$, find the sum $a_{1}+a_{2}+\cdots+a_{99}$
The key to solving this problem lies in recognizing that
$$
a_{n}=\frac{1}{(n+1) \sqrt{n}+n \sqrt{n+1}}=\frac{(n+1) \sqrt{n}-n \sqrt{n+1}}{(n+1)^{2} n-n^{2}(n+1)}=\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}
$$

Substituting into the desired sum, we get
$$
\begin{aligned}
a_{1}+a_{2}+\cdots+a_{99} & =\left(\frac{1}{1}-\frac{1}{\sqrt{2}}\right)+\left(\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}\right)+\cdots+\left(\frac{1}{\sqrt{99}}-\frac{1}{\sqrt{100}}\right) \\
& =\frac{1}{\sqrt{1}}-\frac{1}{\sqrt{100}}=\frac{9}{10}
\end{aligned}
$$

In addition to the five methods of summing sequences mentioned above, there are other methods, such as complex number methods and combinatorial methods, which will be introduced in relevant chapters of this book.

Mastering the summation of sequences can provide significant assistance in solving mathematical problems. Some problems related to sequences can be handled in a refreshing way using sequence summation methods.