I have been trying to do this problem for the past hour and I am getting frustrated with it. Usually when I have to use the integral test the functions are easy to integrate, like 1/x or 1/xln(x), but i know im not supposed to be integrating this huge equation and I am stuck of what to compare it to. Any help is apprecited, thanks!
To study the series \(\displaystyle \displaystyle \sum_{n=0}^{\infty}\, \dfrac{5n\, +\, 8}{(n^2\, +\, 7n\, +\, 10)\, \sqrt[3]{\strut n\, +\, 4\,}},\) we compare with the
integral \(\displaystyle \displaystyle \int_1^{\infty}\, \dfrac{1}{x^p}\, dx,\) where p = _____ .
You are to enter the value \(\displaystyle p\, \in\, \mathbb{Q}\) (as an integer or in\(\displaystyle \frac{a}{b}\) form) such that
\(\displaystyle \displaystyle \lim_{x \rightarrow \infty}\, \left(\dfrac{5x\, +\, 8}{(x^2\, +\, 7x\, +\, 10)\, \sqrt[3]{\strut x\, +\, 4\,}}\right)\, \big/ \left(\dfrac{1}{x^p}\right)\) is a positive number.
Since the integral \(\displaystyle \displaystyle \int_1^{\infty}\, \dfrac{1}{x^p}\, dx\) [ pick one ], we conclude that the series [ pick one ].
I try to compare the big function:
(5n+8) / (n^2 + 7n + 10)*(n+4)^(1/3) <= (5n+8) / (n^2 + 7n + 10) <= (5n+8) / n^2
But the integral of (5x+8) / x^2 dx will diverge because it is of the form of 1/x^p where p = 1. Since (5n+8)/n^2 diverges, it says nothing about the smaller function, which is (5n+8) / (n^2 + 7n + 10)*(n+4)^(3/4). So I am stuck of how to compare the big function to a different function of form 1/x^p because the answer is that this series should converge, which means that we should compare (5n+8) / (n^2 + 7n + 10)*(n+4)^(3/4) to a function that is greater or equal to this, but should converge, so this one will converge by the comparison/integral test.
To study the series \(\displaystyle \displaystyle \sum_{n=0}^{\infty}\, \dfrac{5n\, +\, 8}{(n^2\, +\, 7n\, +\, 10)\, \sqrt[3]{\strut n\, +\, 4\,}},\) we compare with the
integral \(\displaystyle \displaystyle \int_1^{\infty}\, \dfrac{1}{x^p}\, dx,\) where p = _____ .
You are to enter the value \(\displaystyle p\, \in\, \mathbb{Q}\) (as an integer or in\(\displaystyle \frac{a}{b}\) form) such that
\(\displaystyle \displaystyle \lim_{x \rightarrow \infty}\, \left(\dfrac{5x\, +\, 8}{(x^2\, +\, 7x\, +\, 10)\, \sqrt[3]{\strut x\, +\, 4\,}}\right)\, \big/ \left(\dfrac{1}{x^p}\right)\) is a positive number.
Since the integral \(\displaystyle \displaystyle \int_1^{\infty}\, \dfrac{1}{x^p}\, dx\) [ pick one ], we conclude that the series [ pick one ].
I try to compare the big function:
(5n+8) / (n^2 + 7n + 10)*(n+4)^(1/3) <= (5n+8) / (n^2 + 7n + 10) <= (5n+8) / n^2
But the integral of (5x+8) / x^2 dx will diverge because it is of the form of 1/x^p where p = 1. Since (5n+8)/n^2 diverges, it says nothing about the smaller function, which is (5n+8) / (n^2 + 7n + 10)*(n+4)^(3/4). So I am stuck of how to compare the big function to a different function of form 1/x^p because the answer is that this series should converge, which means that we should compare (5n+8) / (n^2 + 7n + 10)*(n+4)^(3/4) to a function that is greater or equal to this, but should converge, so this one will converge by the comparison/integral test.
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