Answer TRUE or FALSE to each of the following statements and justify your answer. Any answer without justification will not be taken into account in the grading. All questions in this exercise are independent.
Consider the sequence $( u _ { n } )$ defined for every non-zero natural number $n$ by $$u _ { n } = \frac { 25 + ( - 1 ) ^ { n } } { n }$$ Statement 1: The sequence $\left( u _ { n } \right)$ is divergent.
Consider the sequence $\left( w _ { n } \right)$ defined for every natural number $n$ by $\left\{ \begin{aligned} w _ { 0 } & = 1 \\ w _ { n + 1 } & = \frac { w _ { n } } { 1 + w _ { n } } \end{aligned} \right.$ It is admitted that for every natural number $n , w _ { n } > 0$. Consider the sequence $( t _ { n } )$ defined for every natural number $n$ by $t _ { n } = \frac { k } { w _ { n } }$ where $k$ is a strictly positive real number. Statement 2: The sequence $\left( t _ { n } \right)$ is a strictly increasing arithmetic sequence.
Consider the sequence $\left( v _ { n } \right)$ defined for every natural number $n$ by $\left\{ \begin{array} { l l l } v _ { 0 } & = 1 \\ v _ { n + 1 } & = & \ln \left( 1 + v _ { n } \right) \end{array} \right.$ It is admitted that for every natural number $n , v _ { n } > 0$. Statement 3: The sequence $( v _ { n } )$ is decreasing.
Consider the sequence $\left( I _ { n } \right)$ defined for every natural number $n$ by $I _ { n } = \int _ { 1 } ^ { \mathrm { e } } [ \ln ( x ) ] ^ { n } \mathrm {~d} x$. Statement 4: For every natural number $n , I _ { n + 1 } = \mathrm { e } - ( n + 1 ) I _ { n }$.
\section*{Exercise 3}
Answer TRUE or FALSE to each of the following statements and justify your answer. Any answer without justification will not be taken into account in the grading. All questions in this exercise are independent.
\begin{enumerate}
\item Consider the sequence $( u _ { n } )$ defined for every non-zero natural number $n$ by
$$u _ { n } = \frac { 25 + ( - 1 ) ^ { n } } { n }$$
Statement 1: The sequence $\left( u _ { n } \right)$ is divergent.
\item Consider the sequence $\left( w _ { n } \right)$ defined for every natural number $n$ by $\left\{ \begin{aligned} w _ { 0 } & = 1 \\ w _ { n + 1 } & = \frac { w _ { n } } { 1 + w _ { n } } \end{aligned} \right.$
It is admitted that for every natural number $n , w _ { n } > 0$.\\
Consider the sequence $( t _ { n } )$ defined for every natural number $n$ by $t _ { n } = \frac { k } { w _ { n } }$ where $k$ is a strictly positive real number.\\
Statement 2: The sequence $\left( t _ { n } \right)$ is a strictly increasing arithmetic sequence.
\item Consider the sequence $\left( v _ { n } \right)$ defined for every natural number $n$ by $\left\{ \begin{array} { l l l } v _ { 0 } & = 1 \\ v _ { n + 1 } & = & \ln \left( 1 + v _ { n } \right) \end{array} \right.$
It is admitted that for every natural number $n , v _ { n } > 0$.\\
Statement 3: The sequence $( v _ { n } )$ is decreasing.
\item Consider the sequence $\left( I _ { n } \right)$ defined for every natural number $n$ by $I _ { n } = \int _ { 1 } ^ { \mathrm { e } } [ \ln ( x ) ] ^ { n } \mathrm {~d} x$.
Statement 4: For every natural number $n , I _ { n + 1 } = \mathrm { e } - ( n + 1 ) I _ { n }$.
\end{enumerate}