A community for students.
Here's the question you clicked on:
 0 viewing
watchmath
 5 years ago
Interesting problem anyone?
Compute
\[\lim_{n\to\infty}\frac{1\cdot 1!+2\cdot 2!+\cdots+n\cdot n!}{(n+1)!}\]
watchmath
 5 years ago
Interesting problem anyone? Compute \[\lim_{n\to\infty}\frac{1\cdot 1!+2\cdot 2!+\cdots+n\cdot n!}{(n+1)!}\]

This Question is Closed

watchmath
 5 years ago
Best ResponseYou've already chosen the best response.0guess is welcome too :D

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0think i even know what it is, but the proof is eluding me so now i have something to think about for the afternoon.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0won't post until i prove it.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Can we not just factor out an n! from each term in the top, and rewrite it as: \[\lim_{n \rightarrow \infty} \frac{n\cdot n![\frac{1}{n\cdot n!} + \frac{2 \cdot 2!}{n\cdot n!} + ... + 1]}{n\cdot n!(1+\frac{1}{n})} = \lim_{n \rightarrow \infty} \frac{\frac{1}{n\cdot n!} + \frac{2 \cdot 2!}{n\cdot n!} + ... + 1}{(1+\frac{1}{n})} = \frac{1}{1} = 1\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0I guess it's not clear that the sum of all those n terms will go to 0 faster than the number of terms are growing but it seems reasonable to me.

watchmath
 5 years ago
Best ResponseYou've already chosen the best response.0Yes, I think it works. We just need to show more rigorously that \[\lim_{n\to\infty}\frac{(nr)\cdot(nr)!}{n\cdot n!}=0\] for all \(0<r<n\).

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0yeah, I'm looking at that now actually ;)

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0actually just showing it works for r=1 will show it for the rest

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0\[(n1)! = \frac{n!}{n} \rightarrow \lim_{n\to\infty}\frac{(n1)\cdot (n1)!}{n\cdot n!} = \lim_{n\to\infty}\frac{(n1)\cdot n!}{n^2 n!} = \lim_{n\to\infty}\frac{(n1)}{n^2} = 0\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0i think it is just \[\frac{(n+1)!1}{(n+1)!}\]yes?

watchmath
 5 years ago
Best ResponseYou've already chosen the best response.0Awesome satellite! :D

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0proof by induction as soon as i figure it out.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0My method doesn't work?

watchmath
 5 years ago
Best ResponseYou've already chosen the best response.0Your method works! Satellite shows that the expression is in fact equal to \(\frac{(n+1)!1}{(n+1)!}\)

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0well actually satellite has not shown anything. i just said it.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0so no credit yet that is for sure.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0i thought it would be simple induction. but i am getting stuck because i keep getting \[n(n+1)!+2(n+1)!\] and i need this to be \[(n+2)!\] which, if it is, is not clear to me.

watchmath
 5 years ago
Best ResponseYou've already chosen the best response.0Are you sure you want a hint. I am afraid it will soil the fun :).

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0ok no hint. perhaps induction is not way to go?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0oh lord i get the dumb guy award

watchmath
 5 years ago
Best ResponseYou've already chosen the best response.0ok, half hint. The summation on the top can be made into a telescoping sum.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0hehe.. sometimes we can't see the forest for the trees. Happens to us all.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0polpak got it. proof by induction done.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0but now i have to think about the telescope.

watchmath
 5 years ago
Best ResponseYou've already chosen the best response.0\((n+1)!1=(n+1)!1!\) :D

watchmath
 5 years ago
Best ResponseYou've already chosen the best response.0I think a hint won't hurt now :D. \(n\cdot n!=((n+1)1)\cdot n!\)
Ask your own question
Sign UpFind more explanations on OpenStudy
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.