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Use the ratio test to determine if the series converges (-1)"(Z) 8n (Zn + 1)Use the root test to determine the series converges775)"Determine using an app...

Question

Use the ratio test to determine if the series converges (-1)"(Z) 8n (Zn + 1)Use the root test to determine the series converges775)"Determine using an appropriate test from the chapter if the series converges E5-] ( 4"9 17. En-1,+9

Use the ratio test to determine if the series converges (-1)"(Z) 8n (Zn + 1) Use the root test to determine the series converges 775)" Determine using an appropriate test from the chapter if the series converges E5-] ( 4"9 17. En-1,+9



Answers

Using the Ratio Test In Exercises $17-38$ , use the Ratio Test to determine the convergence or divergence of the series. If the Ratio Test Test is inconclusive, determine. If the convergence or
divergence of the series using other methods.
$$\sum_{n=0}^{\infty}(n+2)\left(\frac{9}{7}\right)^{n+1}$$

A n is. You go to find to the ends power to to the ends Power plus one and I am just one is ICO to find to the emptiest one's power to to the impasse one class one So it takes a ratio. Hey, EMS has won over a n. It's Niko to fine to the impasse. One's power over to to that contest one. Just one find to the ends. Power to to the ends. That's what this it is. Speaker to five times, two to the last one over two to the chest, one last one. That's if we take the limit. That's intensity infinity. This limits is Nico to five times one, huh? It's because you firewood, too, so this means the limits of pressure is fire over to you. And if I over to it's bigger than strictly. It's bigger than what? So we know this dimension? A I we are languages

And is he going to 90 defense power over into the fist Powered in AM plus one, you go to nine to that place one's power over them, plus one to the fifth power and a and over A and and plus one over a 1,000,000 hits. You can't you're nine then times into the this power and over the last one to fix power. So if the tickle limits of absolute value rebellion over absolutely room is absolutely of aliens, Class one over absolute value of a N issue could soon die, since this quantity is 10 to 1 with ingratitude infinity. So since we know the ratio tend to nine, it is bigger than one using the richer test we know some mentioned 1,000,000,000 diver Cher's.

Okay, so we start off here using the ratio test. We have endless ones. Where times five or six with fire ventless one divided by end square test five or six. The part of end. So this these two terms right here are just dissolved into a five or six. And these two terms right here can be rewritten as so if we divide the top at the rent and denominator in this chair right here by end, we end up with one plus one over and abetted by one. This term goes to zero. So we just end up with five or six, which is less than one to the serious convergence.

OK, so we take the limit as any approaches infinity off to end to end, plus two factorial times end to the power off five divided by two Any factorial multiplied by n plus one to the power of five so we can simplify this down to be. This is equal to to end plus one multiplied by two and plus two multiplied by any over and plus one to the power of five, which is equal to two and plus one times two added plus two. Since this is greater than zero with basically infinity to Siri's that urges.


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