What does the radius of convergence \(R\) measure?

For centre \(2\) and radius \(7\), write the radius condition.

Convert \(|x+1|<3\) to an open interval.

If the ratio-test limit is \(\frac{|x-5|}{4}\), find \(R\).

Find the radius of \(\sum_{n=0}^{\infty}\left(\frac{x}{6}\right)^n\).

Find the radius of \(\sum_{n=0}^{\infty}2^n(x+3)^n\).

Use the ratio test to find \(R\) for \(\sum_{n=1}^{\infty}\frac{(x-1)^n}{n}\).

Find the centre and radius of \(\sum_{n=0}^{\infty}\frac{(x-4)^n}{5^n}\).

Find \(R\) for \(\sum_{n=0}^{\infty}\frac{(x+2)^n}{n!}\).

Find \(R\) for \(\sum_{n=0}^{\infty}n!(x-2)^n\).

If the ratio-test limit is \(8|x+1|\), find the radius and centre.

A student calls \((1,9)\) the radius of convergence. Correct the language if the centre is \(5\).

Find \(R\) for \(\sum_{n=1}^{\infty}\frac{4^n(x-3)^n}{n^2}\).

Find \(R\) for \(\sum_{n=0}^{\infty}\frac{(n+1)(x+2)^n}{3^n}\).

For \(\sum_{n=0}^{\infty}k^n(x-c)^n\) with \(k>0\), find \(R\).

If \(\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|=L>0\), derive \(R\) for \(\sum_{n=0}^{\infty}a_n(x-c)^n\).

Find \(R\) if the ratio-test limit is \(|x|\) times \(\lim_{n\to\infty}\frac{2n+1}{5n-3}\).

Diagnose the error: after \(|x-1|<2\), a student says endpoints must converge because they are exactly distance \(2\).

Why does \(\sum_{n=0}^{\infty}\frac{x^n}{n!}\) have infinite radius?

Compare the radii of \(\sum_{n=0}^{\infty}\frac{x^n}{n!}\) and \(\sum_{n=0}^{\infty}n!x^n\).