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

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

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

Find the radius and interval of convergence of \(\sum_{n=0}^{\infty}\left(\frac{x-2}{3}\right)^n\).

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

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

Use the geometric series to find a power series for \(\frac{1}{(1-x)^2}\), and state where the series is valid.

Use the geometric series to find a power series for \(-\ln(1-x)\), and state where the series is valid before endpoint checks.

Find the coefficient of \(x^4\) in the power series \(e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}\).

A learner says \(\sum_{n=1}^{\infty}\frac{(x-1)^n}{n^2}\) has interval of convergence \((0,2)\) because the radius is \(1\). Explain the missing step and find the correct interval.