# Calculus 2 - Infinite Series

• GreenPrint
In summary, the conversation involved finding the sequence of partial sums and evaluating the limit of the series .9+.09+.009+..., as well as determining the sum of the series. Confusion arose over the correct sequence of partial sums and the limit, with some disagreement over the answer of the second part. Ultimately, it was suggested to recognize the series as .9999... and determine the simple number that it represents.

## Homework Statement

Find the sequence of partial sums {S_n} and evaluate the limit of {S_n} for the following series

.9+.09+.009+...

What is .9+.09+.009+... equal to?

## The Attempt at a Solution

For the first part of the question (find the sequence of partial sums {S_n})
S_n=9(1/10)^n where n >= 1
my teachers assistant marked my answer correct

for the second part of the question (evaluate the limit of {S_n} for the following series .9+.09+.009+...)
I evaluated the limit of S_n by just simply taking the limit of S_n as n goes to infinity
lim n->inf S_n = 9*lim n->inf (1/10)^n = 0
My teachers assistant marked my question wrong and put
S_n = sigma[1,4] 9(1/10)^ character
I can't read what character he put
I don't see how this answer is correct and my answer is wrong. If my answer to finding S_n is correct then why can't I just evaluate the limit as n goes to infinity of S_n to "evaluate the limit of {S_n}? I don't understand what's wrong with my work.

for the third part (What is .9+.09+.009+... equal to?)
.9+.09+.009+... = sigma[n=1,inf] (1/10)^n = 9* (1/10)/(1-1/10) = 9* (1/10)/(9/10) = 9*1/10*10/9 = 1

I don't see how my answer to the second part is wrong. I hope somebody can clear up this confusion for me. Thanks for any help anyone can provide me.

I think your answer to the first part is also wrong. 9*(1/10)^n isn't a partial sum of the series. It's the nth term of the series.

shouldn't the limit of the partial sums be the same as the sum of the series?

Dick said:
I think your answer to the first part is also wrong. 9*(1/10)^n isn't a partial sum of the series. It's the nth term of the series.

I think your first part is okay. They didn't ask for the partial sum, they asked for the sequence of partial sums.

[EDIT] never mind, it does seem wrong.

There are two different sequences here. The first is {.9, .09, .009, ..., 9 * 10-n, ...}
The sequence of partial sums is {.9, .99, .999, ...}

Okay I found the sequence of partial sums to be $$S_n= \frac{9(1+10^{n-1})}{10^n}$$

Then I found the limit of that to be 9/10.

Of course, this may easily be wrong. I'm just making an attempt.

Actually, I think I MUST be wrong about 9/10, since isn't the limit of the sequence of partial sums supposed to equal to the sum of the series?

Oh, I see a problem with my partial sums. Oh well. Your turn GreenPrint. :)

Yes, let's let GreenPrint chime in now.

It might be much simpler to recognize that .9+ .09+ .009+ .0009+ ... is the same as .99999... where the "9" continues for ever. What very simple number is that?