# Calculus 2 - Infinite Series

## 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
my answer was marked correctly

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.

## Answers and Replies

Dick
Science Advisor
Homework Helper
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?

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.

Mark44
Mentor
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. :)

Mark44
Mentor
Yes, let's let GreenPrint chime in now.

HallsofIvy
Science Advisor
Homework Helper
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?