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Numerade Educator



Problem 6 Medium Difficulty

Calculate the first eight terms of the sequence of partial sums correct to four decimal places. Does it appear that the series is convergent or divergent?
$ \displaystyle \sum_{n = 1}^{\infty} \frac {1}{\sqrt [3]{n}} $


The series diverges


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Video Transcript

let's calculate the first eight terms of the sequence correct of four decimal places. So for this instead of going by hand, I'LL just use the calculator. So here I am in Wolfram Alpha And if we look at the first some we only look as one just adding wants her We should have just won over the cube root of one, which is just one. If you want to go for decimals, you could write it that way. Now go to as to if we would write that out. But of course, here we want the approximation. So this time I will Sum of two terms and then on the outside is to make sure that we're rounding off in this five is to give four decibels after the ones digit. So this is why I have five take it for digits to the right of the decimal. So one, seven, nine, three, seven and so on. And we'll keep going all the way up to s ay So for as three this time we'LL add three terms So let's come back to Wolfram Alpha four Hate seven one. So that's two point four aid seven one goingto four decibels and we'LL keep going. We have five more. So now we're on number four. This is the sum of the first four terms. So going to four decibels we have three and then point one one seven zero. Now we'LL go to five terms three and then seven zero one eight That seven zero one day after the three seven Oh one a. That's what we wrote. Now let's go to a six when I'm riding six terms This time we're going upto four point two five to one. That's for a six. That was four point two five two one. Now let's go to seven terms. You know, we're on our last two terms here. Four seven, seven, four nine That's four seven seven four nine. And now we go to the last term by plugging in and equals for the upper bound and eight. And now we have five point two seven four nine. So that was five points. Who? Seven four. Nine. So all of these are our answers. For the first part of the question, we've calculated the first aid terms of the sequence of partial sums. That's s one all the way up to as eight. Now, for the second part of the question, does it appear that the Siri's is convergent or divergent? Well, we start off that one. Then we jumped all the way up almost by point eight. Then we increase almost by two. Then we increased. I'm sorry. I lost out of order here. So first, second. Then we jumped up by about point seven again point six, their point five, another point five. And then increased point five again. So I would say that this is that version and you could actually prove it's diversion by the pee test here. Piers won over three, and that's our final answer.