Root Test

#Calc-2#Math#Series The root test is very similar to the Ratio Test

Given $\sum _{k=1}^{\infty }{a}_k$, it converges absolutely if ${\lim }_{k\to \infty }\sqrt[k]{|a_k|}<1$ It diverges if ${\lim }_{k\to \infty }\sqrt[k]{|a_k|}>1$ The test is inconclusive if ${\lim }_{k\to \infty }\sqrt[k]{|a_k|}=1$

Why? ${\lim }_{k\to \infty }\sqrt[k]{|a_k|}=r$, then this means that as k -> infinity, $|a_k|\approx r^k$ which means $\sum |a_k|=\sum r^k$ which is a geometric series, and if $|r|<1$ it will converge

The Root Test works the best when there is an exponent of k in $a_k$