# Conquering the Conundrum: Simplifying the Sum of n!/(1000^n)

• FrostScYthe
In summary: Thanks for the clarification.In summary, the sum infinitesimally approaches zero as n goes to infinity, but it does not converge.
FrostScYthe
inf
Sum n!/(1000^n)
i = 1

Is that with geometric series... or what?!... all I can think of is splitting n! and the fraction (1/1000)^n, anyway, any help appreciated :)

The first question you need to ask is whether this sum converges or diverges.

Have you studied Big-O notation? Is 1000^n O(n!)? Or visa versa? Don't be fooled by what happens at small values of n. Eventually one of the expressions will overwhelm the other. Which is it? and what happens when it does?

Do the ratio test.

n! would overwhelm 1000^n eventually, I think, if I'm correct... I am guessing it diverges.. but I need to prove this algebraically...
ratio test hmmm I'm not sure how to do this for that?

n!/n! 1
------------ = -------------- agh I don't get anywhere X\
(1000^n)/n! (1000^n)/n!

$$n!$$ would be less than $$1000^{n}$$ for all n. Since $$a_{n} \rightarrow 0$$ as $$n\rightarrow \infty$$, the series converges.

Try the ratio test on this series, I think you're mistaken.

$$\lim_{n\rightarrow \infty} |\frac{(n+1)n!}{1000^{n+1}}\frac{1000^{n}}{n!}| = \frac{n+1}{1000}$$

As $$n\rightarrow \infty$$, $$|\frac{a_{n+1}}{a_{n}}| \rightarrow 0$$. So it converges.

How does (n+1)/1000 approach 0 as n goes to infinity?

I'm trying to reply by quoting your post #5 courtri but the quote juste won't appear.

Anyway, I just want to point out in relation to that post, that given a series $\sum a_n$, it is a nessesary condition for convergence that $a_n\rightarrow 0$ but it is not sufficient! In other words, that some series has a vanishing general term does not means it converges.

However, if a series has a diverging general term, then it diverges. It is a useful criterion for proving a series diverges, but cannot be used as a proof for convergence.

And also in relation to your post #5 (why did you delete it, it was an intructive error that the OP could have beneficiated from). It can be seen in a nice way from the way you wrote the sequence that it diverges:

$$\frac{n!}{1000^n}=\frac{1}{1000}\frac{2}{1000}...\frac{n}{1000}$$

As soon as $n\geq 1000*1000=1000000$ for instance, you can rearange the terms two by two like so (ok it looks ridiculous i admit):

$$\frac{1000000!}{1000^{1000000}}=\left(\frac{1}{1000}\frac{1000000}{1000}\right)\left(\frac{2}{1000}\frac{999999}{1000}\right)...\left(\frac{500000}{1000}\frac{500001}{1000}\right)$$

Everyone's greater than one so it is easy to see that past n=1000000, the general term is always greater than unity, so the limit will not converge to 0, so the series must diverge.

quasar987 said:
Anyway, I just want to point out in relation to that post, that given a series $\sum a_n$, it is a nessesary condition for convergence that $a_n\rightarrow 0$ but it is not sufficient! In other words, that some series has a vanishing general term does not means it converges.

You don't mean vanish. Terms vanish if they are zero, thus you have described a finite sum. Tending to zero is not the same as vanish (being equal to zero).

Cool, I didn't know.

## 1. How do I know which formula to use?

First, read the problem carefully and identify what type of mathematical operation is needed (e.g. addition, subtraction, multiplication, division). Then, refer to your notes or textbook to find the appropriate formula for that operation.

## 2. What should I do if I get stuck?

If you are having trouble solving a sum, take a break and come back to it later with a fresh mind. You can also try breaking down the problem into smaller, more manageable parts. If you are still stuck, ask a classmate or your teacher for help.

## 3. How do I check if my answer is correct?

After solving the sum, double-check your work by plugging your answer back into the original problem and seeing if it makes sense. You can also use a calculator to verify your answer.

## 4. Should I show my work?

Yes, it is important to show your work when solving a sum. This not only helps you keep track of your steps, but it also allows your teacher to see your thought process and provide feedback if needed.

## 5. What if I get a different answer from my classmates?

If you get a different answer from your classmates, it could mean that one of you made a mistake. Go back and check your work or ask your teacher for clarification. It is also possible that there are multiple ways to solve a sum, so your answers may differ but both could be correct.

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