広島大学先進理工系科学研究科情報科学プログラム 2019年8月実施専門科目I 問題2

Author

samparker

, とする。

(1) 実数に対して，を求めよ。

(2) とするとき，

を求めよ。

(3) が有限な値に収束する実数の範囲を定めよ。

Let , where .

(1) Calculate the integral for a real number .

(2) Find the limit

when .

(3) Determine the range of the real number on which the limit converges.

Let

Then the region becomes

The integral becomes

Hence

When , we have

where . Then

To determine the range of for which converges, observe that as becomes large, the integral primarily depends on the behavior of .

Since converges when , and by (2) we know that diverges when . Therefore, .