 Hi, this is a video covering infinite series the root test So here's the root test It says that a series converges Absolutely when the limit as n goes to infinity of the nth root of the absolute value of the series is less than one the series diverges when the limit as n goes to infinity of the nth root of Your absolute value of your series formula is greater than one or has an infinite value The root test is inconclusive When the limit is actually equal to one So where the root test comes in handy is anytime your formula for your series that gives you your terms has n in The exponent that's when the root test is useful most of the time So in our first example will determine whether the series converges or diverges to give me The series n equals one two infinity of e to the two n over n to the n power notice that The variable n is in the exponents of Both the base e and the base n so you have n in the power So first we're going to calculate the nth root of the square root of the absolute value of our series formula a sub n so This gives me the nth root of the absolute value of e to the two n over n to the n power So you'll notice that e to the two n and n to the n power are always going to be positive quantities So there's no need to use absolute value anymore You'll also notice that taking the nth root of something is like raising it to the one over n power Finally, if you distribute the one over n and multiply it by the two n If you take the one over n and distribute it and multiply it by the exponent n You get e to the two n over n over n to the n over n power So perhaps where the nice part of this comes into play is when you do finally simplify and you get e squared over n So that's Part of the root test that's what we'll be taking the limit of as n goes to infinity So take the result we just obtained that e squared over n and we are actually going to go in and calculate the limit as n goes to infinity All right, so the limit as n goes to the infinity of e squared over n while you're taking e squared That's a numeric value. That's a non-zero number and you're dividing it by infinity So remember our example we use all the time. You just take a few dollars and divide it amongst infinitely many people That's practically nothing for everybody So you get zero And that is less than one By the root test The series converges absolutely One more example They give me the series n equals one to infinity of n minus two over five n plus one and all of this is raised to the n power So you see that power of n there? That's a good indication that perhaps the root test would work for us So first let's calculate the nth root of The absolute value of a sub n our formula for our series here So you have the nth root of the absolute value of our series formula a sub n And notice that when you take the nth root of something raised to the n power So nth root of something raised to the n power the n power and the nth root cancel each other out So your left strictly was the absolute value of n minus two over five n plus one. We have to leave the absolute value Because for instance when n is one Because n can be anything from one to infinity when n is one you get a negative value for this fraction here One minus two is negative one. So keep the absolute value That's about as much as we can simplify for now So now we'll actually calculate the limit as n goes to infinity of our quantity We just found so that's the limit as n goes to infinity of the absolute value of n minus two over five n plus one So you consider the most dominating or highest power term on the top which is n And the most dominating or highest power term on the bottom which is five n When you simplify simplify finally you get the limit As n goes to infinity of one-fifth Which is just one-fifth. It's less than one. So by the root test The series converges absolutely So that's all we have for you today. I hope you enjoyed learning about the root test