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) $n + k + 1$