Does this sequence have a limit?

Question

5/4
51/40
511/400
5111/4000
51111/40000

It's a pretty simple sequence, but I'm not even sure how else to express it without listing its terms. Does it approach a value, or increase without bound?

Answer 1

I don't have a nice mathematical solution, but as I look at the first five terms,

1.25
1.275
1.2775
1.27775
1.277775

my guess is that it DOES approach a limit of 1.2777777777777777777 or 1.28

Answer 2

The numerator of the nth term in the sequence is 5*10^(n-1) + [k=0, n-2]∑10^k, and the denominator is 4*10^(n-1). We therefore wish to find:

[n→∞]lim (5*10^(n-1) + [k=0, n-2]∑10^k) / (4*10^(n-1))

First we use the formula for the sum of a geometric series:

[n→∞]lim (5*10^(n-1) + (1-10^(n-1))/(1-10)) / (4*10^(n-1))

Simplifying:

[n→∞]lim (5*10^(n-1) + (10^(n-1)-1)/9) / (4*10^(n-1))
[n→∞]lim (45*10^(n-1) + (10^(n-1)-1)) / (36*10^(n-1))
[n→∞]lim (46*10^(n-1) - 1)/(36*10^(n-1))
[n→∞]lim 46/36 - 1/(36*10^(n-1))

And this obviously converges to 46/36, which is 23/18. This is equivalent to the second poster's guess of 1.277777..., although 1.277777... ≠ 1.28, as (s)he incorrectly stated.

Answer 3

In my opinion, it appears to 'increase without bound'