What is the radius of convergence of the power series [math: x+2x2+3x3+⋯]x+2x^{2}+3x^{3}+\cdots?

Question

BlackTom AI · Accepted Answer

The problem asks for the radius of convergence of the power series formed by the terms x + 2x^2 + 3x^3 + ... which can be written as sum_{n=1}^∞ n x^n.

To analyze convergence, consider the general term a_n = n x^n and apply the ratio test: 
|a_{n+1}/a_n| = |(n+1)x^{n+1} / (n x^n)| = ((n+1)/n) |x|.
As n → ∞, (n+1)/n → 1, so the limit is |x|. The ratio test then says the series converges if this limit is < 1, i.e., if |x| < 1, and diverges if |x| > 1.

Next, examine the boundary points |x| = 1.
- If x = 1, the series becomes sum_{n=1}^∞ n, which diverges.
- If x = -1, the series becomes sum_{n=1}^∞ n(-1)^n, whose terms do not tend to 0, so it diverges as well.

Therefore, the series converges for all x with |x| < 1 and diverges for |x| ≥ 1. In the language of power series, the radius of convergence is 1 because the interval of convergence extends from -1 to 1 but does not include the boundary points, and the radius is the distance from the center (0) to the boundary of the convergence region.

Put succinctly: the radius of convergence is 1.

What is the radius of convergence of the power series [math: x+2x2+3x3+⋯]x+2x^{2}+3x^{3}+\cdots?简答题

类似问题

Suppose the power series \(\displaystyle \sum_{n=1}^{\infty}a_nx^n\) has radius of convergence 4, then the series \(\displaystyle \sum_{n=1}^{\infty}a_n2^n\)

Suppose the power series \(\displaystyle \sum_{n=1}^{\infty}a_nx^n\) converges at \(x=4\) and \(x=-3\), at which of the following values of \(x\) is the series definitely convergent (select one or more)?

Suppose the power series \(\displaystyle \sum_{n=1}^{\infty}a_nx^n\) has radius of convergence 4, then the series \(\displaystyle \sum_{n=1}^{\infty}a_n2^n\)

Question text What is the radius of convergence of the power series [math: ∑n=1∞(n+2)(n+3)x2nn24n]? Answer: [input] Check Question 2

What's the radius of convergence of the power series [math: ∑n=1∞xnn⋅2n?]\sum_{n=1}^{\infty}\frac{x^n}{n\cdot 2^{n}}?

What's the radius of convergence of the power series \(\sum_{n=1}^\infty n! x^n?\)

What is the radius of convergence of the power series \(x+2x^{2}+3x^{3}+\cdots\)?

What's the radius of convergence of the power series \[\sum_{n=1}^{\infty}\frac{x^n}{n\cdot 2^{n}}?\]

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单