Does the Series Sum of a_n Converge When Defined by Digit Restrictions?

[hy23]

Homework Statement

The sequence a_n = 0, if n contains the digit 9
a_n = 1/n. if n does not contain the digit 9

does the series[tex]\sum[/tex] a_n converge?

Homework Equations

The Attempt at a Solution

I have this idea to separate this series into two subseries - the harmonic and the series of all 0's, and maybe calculate the sum of the limit of the two series. My idea doesn't seem logical though because for one, the harmonic has no limit, and defining the subseries is a problem in itself since the occurence of 9's is a pattern hard to define.

One other way I've been told is to count the number of integers that don't have the digit 9 from 1-10, then from 1-100, then from 1-1000, and then take the ratio of this number to all the numbers in the said interval
so from 1-10, there is 9 numbers without 9, the ratio is 9/10
from 1-100, there is 81, the ratio is 81/100
from 1-1000, there should be 81x9, the ratio is 729/1000
so we can write a geometric series a_n= 9ⁿ/10ⁿ, if n=1 represents 1-10, etc.
What I'm astounded by is that the limit of this geometric sequence is 0, so that means at the limit, every number will have a digit 9, and thus the original sequence has a limit of 0 also and it converges?

 

[mighty2000]

I would first try to understand the problem at hand and gather all the necessary information. In this case, the given sequence is defined in two parts - one for numbers that contain the digit 9 and one for those that do not. The first step would be to analyze the behavior of each part individually.

For the first part, we can see that the sequence is simply a constant value of 0. Therefore, the sum of this part would be 0.

For the second part, we have a harmonic series, which is known to diverge. However, this harmonic series only contains numbers that do not have the digit 9. So, we can say that the sum of this part is actually the sum of all the numbers that do not have the digit 9.

Next, we can use the method suggested in the forum post to calculate the ratio of numbers without the digit 9 to all numbers in a given interval. As the interval increases, the ratio approaches 0, meaning that at the limit, all numbers will have the digit 9. This also means that the sum of the harmonic series will also approach 0.

Putting these two parts together, we can say that the sum of the entire sequence an is 0 + 0 = 0. Therefore, the series converges to 0.