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Use the Ratio Test to determine if the series converges. 8k2 2 k71 7k Select the correct choice below and fll in the answer box to complete your choice 0A: The seri...

Question

Use the Ratio Test to determine if the series converges. 8k2 2 k71 7k Select the correct choice below and fll in the answer box to complete your choice 0A: The series diverges because r =The series converges becauseThe Ratio Test is inconclusive because

Use the Ratio Test to determine if the series converges. 8k2 2 k71 7k Select the correct choice below and fll in the answer box to complete your choice 0A: The series diverges because r = The series converges because The Ratio Test is inconclusive because



Answers

Use the Ratio Test to determine whether each series converges or diverges.
$\sum_{k=1}^{\infty} \frac{k}{(2 k-2) !}$

Should test to determine at this series urges or to purchase state the limited scale approaches infinity campus one tutorial over this. One squared. It's K plus two squared. Find out by K squared times. K plus one squared. Okay. Okay. Factorial. Yeah. So even though there's like plus ones on all these here at these are still going to be K to the power for is the highest degree in top to bottom. And the key factories will cancel out the announced with K plus one on the top. It's the limit. As key approaches, infinity should go to infinity. Therefore, this series, the apparatus.

To determine if this series converges or diverges, take the limited K precious infinity of K plus one. Landscape lys 3/3 to the K plus one step by 32 K over K times. Key two. Okay, so Get that simplified to 1/3 here because these two end up going to under one when you take the limit, escape purchase infinity. And then the three case cancels were left with one third, So that is definitely less than one means that this series converges.

Threes Unequalled, 1 to Infinity. All factorial and Squire upon factorial to end. So here in discussion we have to use the restroom has to decide whether the series converges all diverges. Let a. So an equal to trees, Unequal to 1 to infinity of factorial and Squire upon factorial to win. So here in ratio test threes is of and support. The sequence of ratio has a limit ratio is A sub n plus one upon E. Sub N. So here you can see that limit and approaches infinity of Marcellus a sub and plus one upon murillo's A. So and equal to L. So here we know that E. So and plus one equal to trees An equal to 1 to infinity or Factorial and Plus one square upon factorial two n plus two. So we put that value here and we get al equal to limit and approaches infinity of factorial and plus one Squire upon factorial two n plus two time. Factorial to end upon factorial and Squire. So here we know that factorial and plus one square equal square off and plus one time it's quite a factorial and and we know that factorial to n plus two equal to to n plus two time to end plus one time factorial to win. So now we put that value here and we get al equal to limit and approaches infinity of it's quite off and plus one came the square of factorial and upon two times. And plus one time to when plus one time factorial do N time factorial to end upon Squirtle factorial. And so here we simplify this and we get limit and approaches infinity of and plus one upon full time N plus one plus two. Sorry plus two, nobody white denominator and numerator by. And so here we get all equal to limit and approaches infinity of one plus 1 upon and upon four plus two upon. And so here and approaches infinity infinity. We can perform the expedition. So here we get AL equal to one upon four. So here we know that in racial rest if L less than 2 1, then threes a. So bad converges. So here you can see that AL equal to one upon four. So here it is less than 21 so our threes is convergence. So here sees an equal to one to infinitely off, square off factorial and upon factorial to end converges. So it is our final answer.

Test to determine this serious converges or diverges? The limit has K approaches infinity. Okay. Plus one. Mhm. Two, three to the K plus one. All that over K times to over three. Okay. All right. So we simplify this here. This is going to give us These cancel here. The US with 2/3. Come on here. And the K plus one of the K. As king goes to infinity, goes to one and one. So we have left with 2/3. She's definitely less than one. So this serious coverages.


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