Determining absolute vs conditional convergence using the root test
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|}???
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|}???
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.