京都大学情報学研究科知能情報学専攻 2024年8月実施情報学基礎 F1-2

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In the questions below, denotes the natural logarithm of , and denotes Napier's constant (the base of the natural logarithm).

Answer the following questions. Derivations must be clearly shown.

(1) Let be a positive integer. Compute the -th derivative of .

(2) Compute the following limit.

Compute the volume common to a sphere and a cylinder , . Derivation must be clearly shown.

Answer the following questions.

(1) Show that the inequality

holds for any positive integer .

(2) Show that is an irrational number using the inequality in (1).