Optimized "Homework Solutions for Hard Limits

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In summary, for the first question, the limit as n approaches infinity of ((n+1)^5-(n-1)^5)/n^4 simplifies to 2, despite Wolframalpha giving a different answer.For the second question, the limit as n approaches infinity of (n!)^2/(2n)! evaluates to 0, not infinity as originally guessed, because the terms in the expansion of n! on the top and (2n)! on the bottom cancel each other out, leaving a limit of 0.
  • #1
freshman2013
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Homework Statement


1. lim as n approaches infinity of ((n+1)^5-(n-1)^5)/n^4
2. lim as n to infinity (n!)^2/(2n)!

Homework Equations


The Attempt at a Solution


1.I split it up, got (((n+1)/n)^4)*(n+1)-(((n-1)/n)^4)*(n-1). I try to simplify that down to (n+1)-(n-1) and got 2 as my answer, since the other portion of the limit evaluates to 1. Wolframalpha, however, gave me 10 as the answer.

2. I never seen a problem involving limits with just factorials, so I just guessed that n factorial squared grows faaster than (2n)! so the answer is infinity but wolframalpha says the answer is 0.
 
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  • #2
freshman2013 said:
I try to simplify that down to (n+1)-(n-1)
How did you simplify it to that?
Just expand (n+1)5 etc. by the binomial theorem.
2. I never seen a problem involving limits with just factorials, so I just guessed that n factorial squared grows faaster than (2n)! so the answer is infinity but wolframalpha says the answer is 0.
In the expansions of each n! on the top and (2n)! on the bottom, do you see which terms will cancel? What will that leave?
 
  • #3
1. Expand ## (n + 1)^5 ## and ## (n - 1)^5 ##.

2. You guessed wrong. But you don't need to guess - look at the ratio of the (n + 1)th term to the nth term, this is always a good place to start.
 

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