4. $\frac{1}{2 \sqrt{ } 1}+1 \sqrt{2}+\frac{1}{3 \sqrt{2}+2 \sqrt{3}}$ $+\cdots+\frac{1}{100 \sqrt{99}+99 \sqrt{100}}$ The value is ( ).
(A) $\frac{3}{4}$.
(B) $\frac{9}{10}$.
(C) 1 .
(D) $\sqrt{2}$.
(E) None of the above.
4. $\sum_{n=1}^{n} \frac{1}{(n+1)} \sqrt{n}+n \sqrt{n+1}$
$$
\begin{array}{l}
=\sum_{n=1}^{\infty} \frac{(n+1) \sqrt{n}-n \sqrt{n+1}}{(n+1) n} \\
=\sum_{n=1}^{\infty}\left(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\right) \\
=1-\frac{1}{\sqrt{100}}=\frac{9}{10} .
\end{array}
$$

Therefore, the correct choice is (B).