Let $k \in \mathbb { N } ^ { * }$. We denote by $c _ { k }$ the positive real number such that: $c _ { k } \int _ { 0 } ^ { 2 \pi } \left( \frac { 1 + \cos t } { 2 } \right) ^ { k } d t = 1$, and for every real $t$ we set $R _ { k } ( t ) = c _ { k } \left( \frac { 1 + \cos t } { 2 } \right) ^ { k }$. Let $\epsilon \in ] 0 , \pi [$. We set: $d _ { k } ( \epsilon ) = \sup _ { t \in [ \epsilon , 2 \pi - \epsilon ] } R _ { k } ( t )$. Prove then that $$\lim _ { k \rightarrow + \infty } d _ { k } ( \epsilon ) = 0$$
Let $k \in \mathbb { N } ^ { * }$. We denote by $c _ { k }$ the positive real number such that: $c _ { k } \int _ { 0 } ^ { 2 \pi } \left( \frac { 1 + \cos t } { 2 } \right) ^ { k } d t = 1$, and for every real $t$ we set $R _ { k } ( t ) = c _ { k } \left( \frac { 1 + \cos t } { 2 } \right) ^ { k }$.
Let $\epsilon \in ] 0 , \pi [$. We set: $d _ { k } ( \epsilon ) = \sup _ { t \in [ \epsilon , 2 \pi - \epsilon ] } R _ { k } ( t )$. Prove then that
$$\lim _ { k \rightarrow + \infty } d _ { k } ( \epsilon ) = 0$$