Determine the radius of convergence and the interval of convergence for each power series. ∑n=0∞n(x−1)n

Question

Determine the radius of convergence and the interval of convergence for each power series.  ∑n=0∞n(x−1)n

comentezq · Accepted Answer

1) Radius of convergence in power series: Radius of convergence in power series is the radius of largest disk in which the series converges. The given series is P(x)=∑n=0∞n(x−1)n and cn=n The formula for radius of convergence is 1R=limn→∞|cm+1cm| 1R=limn→∞|m+1m| 1R=limn→∞|m+1m| 1R=limn→∞|mm+1m| 1R=limn→∞|1+1m| Putting the limit. 1R=1 {∵1∞=0} Radius od convergence R=1 For interval of convergence: If R is finite and non zero the power series is convergent inside the interval |x−a|&amp;lt;R   ⇒|(x−1)|&amp;lt;1 ⇒−1&amp;lt;x−1&amp;lt;1 ⇒0&amp;lt;x&amp;lt;2 now checking end point at x=0,∑n=0∞n(−1)n at x=2,∑n→0∞n which is P series P=−12 so that it is convergence. hence interval of convergence is [0,2]

Jeffrey Jordon · Accepted Answer

Answer is given below (on video)

Determine the radius of convergence and the interval of convergence for each power series. sum_{n=0}^inftysqrt{n}(x-1)^n

Answered question

Answer & Explanation

New Questions in Calculus 2