In this problem we have to find interval of convergence for the given in finite series. So first thing that we can note here, this whole term represents any term of the series, let us say anyth term is represented by B N. So bien is six X. Raised to the power and divided by fifth root of and therefore and plus one term will be we can we have to replace and by end plus one. So this is going to be six times X raised to the power and plus one divided by fifth root of and plus one. No, we will perform raise your taste here. So from ratio taste what raise your taste says? We have to find limit off absolute value of be in divided by B N plus one at an approaches to infinity. If this value is less than one then the city's converse's. And at the end we would have to check the value at the end points. So first we will calculate limit of the function here and then we will calculate the interval. So what is going to be limited? So limit as an approaches to infinity models of being we will put its value six x rays to the power and divided by fifth root of and and what is B N plus one since it is in denominator. So we will reverse the a numerator and denominator term of be off and plus one. So it will be fifth the root of and plus one divided by six times X raised to the power and plus one. Now a few terms will get canceled here and we'll get canceled. Now we can write X raised to the power and plus one X raised to the power and plus one as X in X raised to the power end into X raised to the power one. So simply X. Now this X raised to the power and will get cancelled with the numerator term here. So the left terms will be limit of and approaches to infinity. The numerator terms will be fifth root of n plus one divided by fifth root of and into once upon X. So limit of and approaches to infinity. We can combine these two terms, fifth, the root of endless one upon and into one by X. We can take an comin from the numerator. So limit of this function will be as an approaches to infinity. Fifth wrote off and in 21 plus one by N divided by N into one upon X. Now as here the animal gets canceled now as the end approaches to infinity so this term will approach to zero. Therefore limit value of this entire function will be one therefore, if you perform limit here. So this function reduces to models of one upon X. Now for the conversions we know that the value of the limits should be less than one. Therefore model Fund by X should be less than one. Now from here we get to solutions first solution is minus one by X minus one way X is greater than one and second solution is one by X is less than one, so if minus one where X is greater than one, then X will be less than minus one. And from here we will get X is greater than one. Therefore the interval of convergence is going to be X belongs to minus infinity to minus one union, one to infinity. But we have to check the endpoints here at minus one and one. So accordingly we will modify this result. No, if we dig if exit the calls to one so what will be our cities? So series was energy equals to eight to infinity six and two x rays to the power and divided by fifth root of And so if you put the value of x rays to the power in here. So if you put value of X equals to one here. So this will reduce to sigma. Energy costs 218 to infinity six and 21 raised to the power N divided by fifth root of. And now we know that power of power any number to the power of one will give one. Therefore this will be shake him off and equals to eight to infinity six divided by fifth root of and no we can use we can use divergence taste here. So what divergence taste? It says if you perform divergence test and we take the limit. If we take the limit of the any term of the function here. Six divided by fifth root of n. As N approaches to infinity. So we see that this is a close to zero. It means which is less than one. So if it is, if the limit of the function at any and approaches to infinity is less than one, this implies that the function is conversion. Therefore X equals to one is going to be convergent. So we will use equality at X equals to one. So our so our interval will modify too. X belongs to one to infinity. Now we will check the scene for X equals two minus one. So if exes equals two minus one, so at X equals two minus one. Our cities will modify us. What was our series? It was sigma from any close to eight to infinity. Six x rays to the power and upon fifth root of and that is equals two, six and two minus one raised to the power and divided by fifth root of and and limit as N approaches to infinity. Now if we now if we go for divergence taste then limit as an approaches to infinity of the term that is minus one raised to the power N into six, divided by fifth root of. And we observe here that here the term present is minus one raised to the power end, which is an oscillating limit, which will give an oscillating limit. This implies that limit does not exist. Limit does not exist if limited does not exist. This implies that a taxi cost to minus one. The cities will not be convergent. Therefore that there is the engine of the range of convergence will be X belongs to minus infinity to minus one. There will be an open bracket at taxes equals two minus one union, one to infinity, and one will be included in our solution. Therefore, there will be closed interpreter, so this is going to be our final answer.