Let us define a sequence $\left\{ S _ { n } \right\}$ as

$$S _ { n } = \sum _ { k = 1 } ^ { n } \frac { 1 } { \sqrt { k } } \quad ( n = 1,2,3 , \cdots ) .$$

We are to find the following two limits:

$$\begin{aligned}
& \lim _ { n \rightarrow \infty } S _ { n } , \\
& \lim _ { n \rightarrow \infty } \frac { S _ { 2 n } - S _ { n } } { \sqrt { n } } .
\end{aligned}$$

(1) For each of A $\sim$ I in the following sentences, choose the appropriate answer from among (0) $\sim$ (9) at the bottom of this page.

Let us find $\lim _ { n \rightarrow \infty } S _ { n }$. Look at the function $y = \frac { 1 } { \sqrt { x } }$. We have

$$y ^ { \prime } = - \frac { \mathbf { A } } { 2 \sqrt { x ^ { \mathbf { B} } } } ,$$

and hence this function $y$ is $\square$ C . So, considering each interval $k \leqq x \leqq k + 1 ( k = 1,2 , \cdots , n )$, we obtain

$$\frac { 1 } { \sqrt { k } } \mathbf { D } \int _ { k } ^ { k + 1 } \frac { 1 } { \sqrt { x } } d x .$$

When we separately add the left-hand sides and the right-hand sides of this expression from $k = 1$ to $k = n$, we have

$$S _ { n } \mathbf { E } \int _ { \mathbf { F } } ^ { \mathbf { G } } \frac { 1 } { \sqrt { x } } d x = \mathbf { H } ( \sqrt { \square \mathbf { G } } - 1 )$$

and finally

$$\lim _ { n \rightarrow \infty } S _ { n } = \infty .$$

Choices:\\
(0) $\infty$\\
(1) 1\\
(2) 2\\
(3) 3\\
(4) $n$\\
(5) $n + 1$\\
(6) $<$\\
(7) $>$\\
(8) monotonically increasing\\
(9) monotonically decreasing

(2) For each of $\square$ J $\sim$ $\square$ P in the following, choose the appropriate answer from among (0) $\sim$ (9) below.

Let us find $\lim _ { n \rightarrow \infty } \frac { S _ { 2 n } - S _ { n } } { \sqrt { n } }$. Since

$$S _ { 2 n } - S _ { n } = \sum _ { k = 1 } ^ { n } \frac { 1 } { \sqrt { \mathbf { J } } } ,$$

we have from quadrature (mensuration) by parts that

$$\begin{aligned}
\lim _ { n \rightarrow \infty } \frac { S _ { 2 n } - S _ { n } } { \sqrt { n } } & = \lim _ { n \rightarrow \infty } \frac { 1 } { \mathbf { K } } \sum _ { k = 1 } ^ { n } \frac { 1 } { \sqrt { \mathbf { L } + \frac { k } { n } } } \\
& = \int _ { \mathbf { M } } ^ { \mathbf { N } } \frac { 1 } { \sqrt { 1 + x } } d x \\
& = \mathbf { O } ( \sqrt { \mathbf { P } } - 1 ) .
\end{aligned}$$

Choices:\\
(0) 0\\
(1) 1\\
(2) 2\\
(3) $n - 1$\\
(4) $n$\\
(5) $n + 1$\\
(6) $n - k$\\
(7) $n + k$\\
(8) $n + k - 1$\\
(9) $n + k + 1$