Determining absolute vs conditional convergence using the root test

 
 
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Rules for the root test for convergence

The root test for convergence lets us determine the convergence or divergence of a series ???a_n??? using the limit

???L=\lim_{n\to\infty}\sqrt[n]{|a_n|}???

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The convergence or divergence of the series depends on the value of ???L???.

the series converges absolutely if ???L<1???.

the series diverges if ???L>1??? or if ???L??? is infinite.

the test is inconclusive if ???L=1???.

The root test is used most often when our series includes something raised to the ???n???th power.

 
 

Using the root test to determine absolute or conditional convergence


 
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How to apply the root test for convergence

Example

Use the root test to say whether the series converges or diverges.

???\sum^{\infty}_{n=1}\frac{6^n}{(n+2)^n}???

To use the root test, we need to solve for the limit

???L=\lim_{n\to\infty}\sqrt[n]{|a_n|}???

and then evaluate the value of ???L???.

???L=\lim_{n\to\infty}\sqrt[n]{\left|\frac{6^n}{(n+2)^n}\right|}???

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The convergence or divergence of the series depends on the value of L.

We can drop the absolute value bars since all of our terms will be positive.

???L=\lim_{n\to\infty}\sqrt[n]{\frac{6^n}{(n+2)^n}}???

???L=\lim_{n\to\infty}\left[\frac{6^n}{(n+2)^n}\right]^{\frac{1}{n}}???

???L=\lim_{n\to\infty}\left[\left(\frac{6}{n+2}\right)^n\right]^{\frac{1}{n}}???

???L=\lim_{n\to\infty}\left(\frac{6}{n+2}\right)^\frac{n}{n}???

???L=\lim_{n\to\infty}\frac{6}{n+2}???

???L=\frac{6}{\infty+2}???

???L=\frac{6}{\infty}???

???L=0???

Since ???L<1???, we can say that the original series ???a_n??? converges absolutely.

 
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