Question

For \(n>0,\) let \(R>0\) and \(c_{n}>0 .\) Prove that if the interval of convergence of the series \(\sum_{n=0}^{\infty} c_{n}\left(x-x_{0}\right)^{n}\) is \(\left(x_{0}-R, x_{0}+R\right]\). then the series converges conditionally at \(x_{0}+R\).

Short answer

Given the conditions of the problem, it can be proven that the power series \(\sum_{n=0}^{\infty} c_{n}(x-x_{0})^{n}\) converges conditionally at the point \(x_{0}+R\). The series is proven to be convergent at this point, but it is not absolutely convergent, thereby satisfying the definition of conditional convergence.

Step-by-step solution

Conditional Convergence

For the given series \(\sum_{n=0}^{\infty} c_{n}(x-x_{0})^{n}\), let's consider the point \(x_{0}+R\). At this point, the series becomes, \(\sum_{n=0}^{\infty} c_{n}R^{n}\).

Test for Convergence

To find if the series converges conditionally, we first test for convergence. According to the interval of convergence we are given, our series \(\sum_{n=0}^{\infty} c_{n}R^{n}\) converges at \(x_{0}+R.\) Therefore, using this information, we can say that our series is indeed convergent at this point.

Test for Absolute Convergence

If a series is absolutely convergent, then it is also convergent. If our series was absolutely convergent at \(x_{0}+R,\) the interval of convergence would have been \([x_{0}-R, x_{0}+R]\) instead of \((x_{0}-R, x_{0}+R].\) Given that the interval of convergence excludes the point \(x_{0}-R,\) we can conclude that the series \(\sum_{n=0}^{\infty} c_{n}R^{n}\) is not absolutely convergent at \(x_{0}+R.\)