Radius of Convergence: Where Does a Power Series Converge?
Read this topicExplanations 3
- In the previous topic, we looked at convergence of power series ∑n=0∞cn(x−a)n\sum_{n=0}^\infty c_n(x-a)^n∑n=0∞cn(x−a)n for various values of x∈Rx\in\mathbb{R}x∈R, and observed…
By Panavia Shou, Victor Wang
- We discussed in the previous explanation how to find the radius of convergence of a power series ∑n=0∞cn(x−a)n\sum_{n=0}^\infty c_n(x-a)^n∑n=0∞cn(x−a)n. We did this by applyin…
By Panavia Shou, Moor Xu, Alex Federspiel, Victor Wang
- Given a power series in xxx centered at a point aaa, what can we say about the set of inputs for which the power series converges? Since power series are modeled off of geometri…
By Victor Wang
Problems 6
- S=∑n=0∞(x−42)nS = \sum_{n=0}^\infty \left(\frac{x - 4}{2}\right)^nS=n=0∑∞(2x−4)n What is the radius of convergence of this power series?
- ∑n=0∞5n(x−3)3n\sum_{n=0}^\infty 5^n (x-3)^{3n}n=0∑∞5n(x−3)3n What is the radius of convergence of this series?
- ∑n=0∞n2xn\sum_{n=0}^\infty \frac{n^2}{x^n}n=0∑∞xnn2 What is the interval of convergence for this series?
- Let r,a∈Rr, a \in \mathbb{R}r,a∈R. What is the interval of convergence for the following series? ∑n=0∞rn(x−a)n\sum_{n=0}^\infty r^n(x-a)^nn=0∑∞rn(x−a)n
- ∑n=0∞3−n(x−2)2n\sum_{n=0}^\infty 3^{-n}(x-2)^{2n}n=0∑∞3−n(x−2)2n What is the interval of convergence for this series?
- What is the interval of convergence for the following series? ∑i=0∞(x2x+1)i\sum_{i=0}^\infty \left(\frac{x^2}{x+1}\right)^ii=0∑∞(x+1x2)i