【微分積分】5-1-3 積級数｜問題集
1.次のコーシー積を求めなさい。
(1)\(\displaystyle \sum_{n=0}^{\infty}1, \sum_{n=0}^{\infty}x^n\)
\(\displaystyle c_n=\sum_{k=0}^{n}a_kb_{n-k}\)
\(\displaystyle =\sum_{k=0}^{n}1・x_{n-k}\)
\(\displaystyle =\frac{1-x^{n+1}}{1-x}\)
\(\displaystyle =\sum_{k=0}^{n}1・x_{n-k}\)
\(\displaystyle =\frac{1-x^{n+1}}{1-x}\)
(2)\(\displaystyle \sum_{n=0}^{\infty}\frac{1}{n!}, \sum_{n=0}^{\infty}\frac{1}{n!}\)
\(\displaystyle c_n=\sum_{k=0}^{n}a_kb_{n-k}\)
\(\displaystyle =\sum_{k=0}^{n}\frac{1}{k!}・\frac{1}{(n-k)!}\)
\(\displaystyle =\frac{2^n}{n!}\)
\(\displaystyle =\sum_{k=0}^{n}\frac{1}{k!}・\frac{1}{(n-k)!}\)
\(\displaystyle =\frac{2^n}{n!}\)
次の学習に進もう！