## SUBJECT III

Let the arithmetic progression a1, a2, ..., an, ... with the ratio and the first term strictly positive.

a) Calculate the limit of the sequence (an).

b) Calculate:

$$
\lim _{n \rightarrow \infty} \frac{1}{a_{n}}\left(\frac{a_{1}}{\sqrt{a_{1}}}+\frac{a_{2}}{\sqrt{a_{1}}+\sqrt{a_{2}}}+\cdots+\frac{a_{n}}{\sqrt{a_{1}}+\sqrt{a_{2}}+\cdots+\sqrt{a_{n}}}\right)
$$
## SUBJECT 3 -7p

![](https://cdn.mathpix.com/cropped/2024_06_07_67a5faed1d056830fca0g-3.jpg?height=57&width=1577&top_left_y=243&top_left_x=228)$\qquad$
5p b) We use Cesaro-Stolz and the desired limit is equal to $L=\lim \left(\frac{a_{n+1}}{\sqrt{a_{1}}+\sqrt{a_{2}}+\cdots \cdot \sqrt{a_{n+1}}}\right) /\left(a_{n+1}-a_{n}\right)$

$=\frac{1}{r} \lim \frac{a_{n+1}}{\sqrt{a_{1}}+\sqrt{a_{2}}+\cdots .+\sqrt{a_{n+1}}}$

$2 p$

For the new limit, we apply Cesaro-Stolz again and obtain $L=\frac{1}{r} \lim \frac{a_{n+2}-a_{n+1}}{\sqrt{a_{n+2}}}$ $2 p$

$\mathrm{L}=\lim \frac{1}{\sqrt{a_{n+2}}}=0$