What is an interval of convergence for a power series in \(x\)?

If \(c=1\), \(R=2\), and neither endpoint converges, what is the interval?

If \(c=4\), \(R=3\), and both endpoints converge, write the interval.

Find the centre and radius of the open interval \((-2,6)\).

Find the interval of convergence of \(\sum_{n=0}^{\infty}x^n\).

Find the interval of convergence of \(\sum_{n=1}^{\infty}\frac{x^n}{n}\).

Find the interval of convergence of \(\sum_{n=1}^{\infty}\frac{(x-2)^n}{n^2}\).

Find the interval of convergence of \(\sum_{n=1}^{\infty}\frac{(x+2)^n}{n}\).

Find the interval of convergence of \(\sum_{n=1}^{\infty}\frac{(x-3)^n}{n2^n}\).

Find the interval of convergence of \(\sum_{n=1}^{\infty}\frac{(x+1)^n}{n^3}\).

Why does the ratio test not usually decide endpoints such as \(x=c\pm R\)?

If \(R=\infty\), what is the interval of convergence?

Find the interval of \(\sum_{n=1}^{\infty}\frac{3^n(x-1)^n}{n}\).

Find the interval of \(\sum_{n=1}^{\infty}\frac{n(x+4)^n}{2^n}\).

For \(\sum_{n=1}^{\infty}\frac{(x-c)^n}{n}\), determine the interval in terms of \(c\).

For \(\sum_{n=1}^{\infty}\frac{(x-c)^n}{n^2}\), determine the interval in terms of \(c\).

A series has centre \(0\), radius \(4\), left endpoint convergent, and right endpoint divergent. Write the interval.

Diagnose the error: after finding \(R=3\), a student writes a closed interval without testing endpoints.

Give two same-radius examples with different endpoint behaviour.

Why can endpoint convergence matter in a boundary-value physical model with parameter \(x\)?