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(3 pts) Are the following series convergent or divergent? If convergent_ find the sum .26-#)...

Question

(3 pts) Are the following series convergent or divergent? If convergent_ find the sum .26-#)

(3 pts) Are the following series convergent or divergent? If convergent_ find the sum . 26-#)



Answers

$9-26$ Determine whether the series is convergent or divergent.
$$
\sum_{n=3}^{\infty} n^{-0.00000}
$$

Yeah. For the given dramatic Siri's, we have the first term age ableto three on the commonplace. It can be found out by the Pretender terms minus four. Andi three in the common surgical to minus four by three. Now, as you can see, the magnitude off the common racial you better than one. So when this happens that geometric cities she did divergent, I would just

All right. We want to evaluate the convergence of the series given by the some from an equal three to infinity of three and minus four over n squared minus two N. So this question is testing our ability to use convergence tests to evaluate the series convergence. Specifically, we're going to rely on the integral test to solve. So remember that the integral test states that if the integral from K to infinity, fx dx converges. So too, must the sum from N equals K to infinity. Am. Similarly, if the integral from K to infinity fx dx diverges. So too must that series. So if we convert A. M to the form of X. Dx and take the integral from K to infinity, we can solve this problem. So we have that the integral from zero to infinity of three x minus four over experiments to dx is equal to the integral of three from three to infinity of two over x plus one over x minus two dx. This step here in particular was done by partial sums or rather partial fractions method of integration. So this has anti derivative to l n x plus l n x minus two from three to infinity. We see that plugging in infinity will have both algorithms shoot off to infinity. That means are integral diverges, which means our series must also diverged with the integral test.

Another way to rewrite this. Siris, of course, is one over and to the zero point nine 999 and this matches with the P Siri's and it will converge as long as this is greater than or equal to one. But because it's less than one or less than or equal one, I should say that this is going to be a divergent Siri's by the peace serious test.

One way to think of this problem is comparing it back to one over M squared, which we know what would which we know would converge because of the P Siri's being greater than one that value. And so we have a greater than one value here. So therefore, if this one we know converges and it's greater than ours than that, forces ours, which must be smaller, also have toe converge as well. So by the P Series test and the comparison, we know that this one will be a convergent Siri's.


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