grandes-ecoles 2023 Q6

grandes-ecoles · France · centrale-maths2__official Reduction Formulae Evaluate a Closed-Form Expression Using the Reduction Formula

Using the recurrence relation $K _ { n } = K _ { n + 1 } + \frac { 1 } { 2 n } K _ { n }$ and the fact that $I_n \sim K_n$, deduce a simple equivalent of $I _ { n }$ as $n$ tends to $+ \infty$, where $I _ { n } = \int _ { 0 } ^ { 1 } \frac { 1 } { \left( 1 + t ^ { 2 } \right) ^ { n } } \mathrm {~d} t$.

Using the recurrence relation $K _ { n } = K _ { n + 1 } + \frac { 1 } { 2 n } K _ { n }$ and the fact that $I_n \sim K_n$, deduce a simple equivalent of $I _ { n }$ as $n$ tends to $+ \infty$, where $I _ { n } = \int _ { 0 } ^ { 1 } \frac { 1 } { \left( 1 + t ^ { 2 } \right) ^ { n } } \mathrm {~d} t$.

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