作业帮幂级数求收敛域内的和函数 下面该怎么做啊
来源：学生作业帮 编辑：作业帮 分类：数学作业 时间：2024/07/17 13:52:23
幂级数求收敛域内的和函数 下面该怎么做啊
∑{1 ≤ n} (2n-1)·x^(2n-2)/2^n
= ∑{1 ≤ n} (x^(2n-1)/2^n)'
= (∑{1 ≤ n} x^(2n-1)/2^n)'
= (∑{0 ≤ n} x^(2n+1)/2^(n+1))'
= (x/2·∑{0 ≤ n} x^(2n)/2^n)'
= (x/2·∑{0 ≤ n} (x²/2)^n)'
= (x/2·1/(1-x²/2))'
= (x/(2-x²))'
= (2+x²)/(2-x²)².
代入x = 1得∑{1 ≤ n} (2n-1)/2^n = 3.
= ∑{1 ≤ n} (x^(2n-1)/2^n)'
= (∑{1 ≤ n} x^(2n-1)/2^n)'
= (∑{0 ≤ n} x^(2n+1)/2^(n+1))'
= (x/2·∑{0 ≤ n} x^(2n)/2^n)'
= (x/2·∑{0 ≤ n} (x²/2)^n)'
= (x/2·1/(1-x²/2))'
= (x/(2-x²))'
= (2+x²)/(2-x²)².
代入x = 1得∑{1 ≤ n} (2n-1)/2^n = 3.