広島大学 先進理工系科学研究科 情報科学プログラム 2017年8月実施 専門科目I 問題2

Author

samparker

Description

(1) とする時、関数 の における最大値を求めよ。

(2) 以下の広義積分が において収束することを示せ。

(3) 極限 を求めよ。

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(1) For , find the maximum of the function on .

(2) Show the following improper integral converges on :

(3) Find the limit:

Kai

(1)

Note that when and when . Therefore, the maximum of is .

(2)

Let’s divide the integral in a sum of two terms,

For the first term, since the function is decreasing, it's maximum on the interval is attained at , hence

But for , this last integral converges to .

For the second term, since the exponential grows faster than any polynomial, for every we can take so big that so

which completes the proof.

(3)