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watchmath
 5 years ago
Compute
\[
\lim_{n\to\infty}\frac{5+55+\cdots+\overbrace{55\ldots 5}^{n\text{ digits }}}{10^n}
\]
watchmath
 5 years ago
Compute \[ \lim_{n\to\infty}\frac{5+55+\cdots+\overbrace{55\ldots 5}^{n\text{ digits }}}{10^n} \]

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anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Looking at the top we can factor out a 5. 5(1+11+111+1111+...) =5*(n+10*(n1)+100*(n2)+...) we can write this as a sum \[\sum_{i=0}^{n}10^{i}*(ni)\] I just used maple to compute the sum for me. The result is: \[n*10^{n+1}/9(10^{n+1}/9)*(n+1)+(10/81)*10^{n+1}n/910/81\] dividing out 10^n we get: 10n/910n/910/9+100/81n/(9*10^n)10/(81*10^n) the terms with 10^n in the denominator go to 0 and after simplifying and remembering to multiply by our factor of 5 we get: 50/81

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0let me know what you think

watchmath
 5 years ago
Best ResponseYou've already chosen the best response.0Yes, it seems good! Can we try to figure out the computation so we can avoid using maple. I am sure we can do it by hand.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0ok let me think about it

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0side note, do you study modern algebra? I was looking at your site and saw some good stuff on there.

watchmath
 5 years ago
Best ResponseYou've already chosen the best response.0yes :). If you are willing to contribute there I would be very glad :D. You may ask questions there though :D

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0i am planning on getting a doctorate in algebra so maybe I will start to visit the site. Any way we can split the sum into a geometric series and one that is not quite as easy to compute. \[n*\sum_{i=0}^{n}10^{i}+\sum_{i=0}^{n}i10^{i}\] i can compute the first do you know how to compute the second?

watchmath
 5 years ago
Best ResponseYou've already chosen the best response.0it seems we need to split the second one into several sigmas too. Do you already have some school in your mind?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0im looking at ohio state, university of michigan, university of illinois, university of chicago, kent, case western. i will apply to those i think. Im trying to write the second sum out to see if it can be split nicely.

watchmath
 5 years ago
Best ResponseYou've already chosen the best response.0The are good schools in algebra. Where are you studying right now?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0im getting a master's in math at cleveland state. I will probably start the phd from scratch though.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0well nothing is popping out at me to simplify the summation. What do you think?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0maybe look at it is a derivative

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0i think that would work actually

watchmath
 5 years ago
Best ResponseYou've already chosen the best response.0BTW I am in algebra program too. My research is on coding theory over ring :).

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0cool that is what my current algebra professor studies. I changed my mind again I think looking at it as a derivative we can solve by hand

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0so that is why you came up with that annoying ring question!

watchmath
 5 years ago
Best ResponseYou've already chosen the best response.0welcome to the conversation satellite :D

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0sorry to butt in. i was looking for the fibonacci thread but i cannot find it.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0can you so a search on these threads? i promised a reply

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0derivative of geometric series... \[(n+1)a^{n}*(1/(a1))+a^{n+1}*(1/(a1)^{2})+1/(a1)^{2}\] then multiply by a. and that's the second sum (a=10)

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0so we can do it by hand as well

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0this was the most interesting problem i've seen on this site yet

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0where did you find this problem?

watchmath
 5 years ago
Best ResponseYou've already chosen the best response.0hi rsvitale you were right. Actually I did that derivative problem from some other guy yesterday. I found that \[ \frac{x}{(1x)^2}=\sum_{n=1}^\infty nx^n \] and our sum is of the form \(5\sum_{i=0}^n (ni)10^{(in)}=5\sum_{j=0}^n j 10^{j}\) As \(n\to\infty\) we have \(5\frac{(1/10)}{(1(1/10))^2}=\frac{50}{81}\) which is agree to your calculation.
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