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Use the Ratio Test to determine if the following infinite series converges or not_ OO nV24 24-n n=1...

Question

Use the Ratio Test to determine if the following infinite series converges or not_ OO nV24 24-n n=1

Use the Ratio Test to determine if the following infinite series converges or not_ OO nV24 24-n n=1



Answers

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

Threes an equal to one to infinity of and upon two to the power. And so here in this kitchen we have to use the record, has to decide whether those threes converges or diverges. So here we know that in Rachel rest. So here let is of an equal to squeeze an equal to one to infinity of and upon two to the power. And the sequence of ratio has a limit. So here you can see that limit and approaches infinity of Marcellus. He so and plus one more or less. E sub and equal to help. So here we have to find help and we have to decide whether the threes converges all diverges. So here you can see that E sub and plus one equal to threes and equal to one to infinity or and plus one upon to the power and plus one. So we put here that value and we get limit and approaches infinity of Marcellus and plus one upon two to the power and plus one upon and upon two to the power and now we have to put here more or less now we simplify this and we get limit and approaches infinity of and plus one upon to N. Now here we have to divide denominator and numerator so he by and so here we get limit and approaches infinity of one plus one upon N upon do an upon and so here we get one up on two after using direct substitution and approaches infinity. So here we get one up on two. So here you can see that al equal to one upon to now since the ratio of two large terms is less than one. So here we know that two ratio test l less than 21 then stories and you could do one to infinity and a bond to to depart end converges because here you can see that al equal to one upon too. So it is less than 21 So here threes will be converges. So it is a final answer

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.

Squeeze An equal to 1 to infinity of factorial and time factorial. Angeles, one upon factorial to n. So here in discussion we have to use the ratio task to decide whether the threes converges all diverges. Let E. So an equal to 3s an equal to 1 to infinity of Factorial and time factorial and plus one upon factorial two. And so here we know that in ratio test threes a sub bands of all. The sequence of ratio has a limit. So here you can see that limit and approaches infinity of sequence of ratio which is Modelos. E. So and plus one upon E. So N equal to L. So here we know that a sub N plus one equal to squeeze An equal to 1 to infinity of And plus one factorial for factorial and plus one time factorial and plus two upon factorial To end. Plus two. So here we put that value and here we get al equal to limit and approaches infinity of factorial And plus one time factorial and plus two time. Sorry Upon factorial to end plus two game factorial end. So in fact real too. And upon factorial and time Factorial and Plus one. So here we know that Factorial two n Plus two equal to to end plus two times two n plus one time factorial to end. And we know that Factorial and Plus two equal to and plus two time and plus one time factorial. And so now we put that value here and we simplify this. So you can see that here, AL equal to limit and approaches infinity or and plus two game and plus one time factorial and upon two and plus two time two and plus one time factorial to win time factorial to win upon factory and and here you can see that factorial and plus one cancel out here and now we simplify this so we get al equal to limit and approaches infinity of and plus two upon full time and plus two. No, we have to divide numerator and denominator by and so we get alec will too limit and approaches infinity of one plus two upon and upon four plus true upon. And now here you can see that and approaches infinity so we can perform direct substitution. So here we get al equal to one up on four. So you can see that here to raise your desk. If Less than 2 1, then squeeze a sub N converges. So here you can see that one upon four, less than 21 So we can say three's unequal to want to infinity of Factorial and time factorial and plus one upon factorial to when converges. So it is over for lunch.

Threes an equal to one to infinity of one upon factorial two in. So here in the solution we have to use or ratio test to decide whether the threes converges or diverges. So here let ace of an equal to three's, an equal to one to infinity or one upon factorial to end. So here we know that in the ratio test threes a sub N, suppose the sequence of ratio has a limit limit and approaches infinity. Oh Morella's E. So and plus one upon Morella's a sub N equal to L. So here we know that a sub and plus one equal to threes an equal to one to infinity or one upon factorial two, N plus one. So here we know that E so N plus one equal to one Upon factorial two, N plus two. Now we put that value here and we get limit and approaches infinity of one. Upon factorial two and plus two time factorial to win. So here we know that factorial two, N plus two, You can write like that, do N plus two times do and plus one time factorial do N. So we put here and we get limit and approaches infinity of factorial to win a pawn do N plus two times two N plus one time factorial to win. So now we simplify this and we get limit and approaches infinity of one upon do N plus two time two, N plus one. No, we simplify this and we get limit and approaches infinity of one upon full time and Squire plus six times. And plus to now we have to perform direct substitution and approaches infinity. So here we get al equal to zero. And here we know that in ratio test if l less than one then three's a sub and converges. So here you can see that al equal to zero. So squeeze unequal to one to infinity oh one upon two. And factorial two and converges so it is all foreign lands


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