Question

# 8 Σ n=1 (x-4)-1 3 n...

Calculate the interval and radius of convergence given (see image) Answer the following in order: 1. Limit expression for the ratio test: 2. Evaluation of convergence at the left and right endpoints(Interval of convergence: 3. Radius of convergence:
Transcribed: 18 Σ n = 1 η (x-4)-1 39

Given expression is

$\sum _{n=1}^{\infty }\frac{{\left(x-4\right)}^{n-1}}{{3}^{n}n}$

Let

${u}_{n}=\frac{{\left(x-4\right)}^{n-1}}{{3}^{n}n}$

then

${u}_{n+1}=\frac{{\left(x-4\right)}^{n}}{{3}^{n+1}\left(n+1\right)}$

Then

$\underset{n\to \infty }{\mathrm{lim}}\frac{{u}_{n+1}}{{u}_{n}}\phantom{\rule{0ex}{0ex}}=\underset{n\to \infty }{\mathrm{lim}}\frac{\left(x-4\right)}{{3}^{n+1}\left(n+1\right)}×\frac{{3}^{n}n}{{\left(x-4\right)}^{n-1}}\phantom{\rule{0ex}{0ex}}=\underset{n\to \infty }{\mathrm{lim}}\frac{n}{\left(n+1\right)}×\left|\frac{\left(x-4\right)}{3}\right|\phantom{\rule{0ex}{0ex}}=\left|\frac{\left(x-4\right)}{3}\right|$

For convergence

So interval of convergence is $\mathbf{\left[}\mathbf{1}\mathbf{,}\mathbf{7}\mathbf{\right)}$

and radius of convergence $\mathbf{3}$.

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