# Thread: does this diverge or converge?

1. ## does this diverge or converge?

Hi,

I have a function f(x) and I need to find this limit:

$\lim_{x\to \infty }\frac{f(x)}{(x+1)^3}=?$

(Actually, in my case f(x) is a sequence i.e. what I am calling f(x) is actually $\sum_{n=0}^{x}a_n$ I don't know if this makes a big difference or not.)

Can someone point in the right direction please?

Thanks

2. Originally Posted by rainer
Hi,

I have a function f(x) and I need to find this limit:

$\lim_{x\to \infty }\frac{f(x)}{(x+1)^3}=?$

(Actually, in my case f(x) is a sequence i.e. what I am calling f(x) is actually $\sum_{n=0}^{x}a_n$ I don't know if this makes a big difference or not.)

Can someone point in the right direction please?

Thanks

This is a weird question: we have to know what $f(x)$ or, at least, its behavior when $x\rightarrow \infty$ to have some chance to answer..

Tonio

3. Originally Posted by rainer
Hi,

I have a function f(x) and I need to find this limit:

$\lim_{x\to \infty }\frac{f(x)}{(x+1)^3}=?$

(Actually, in my case f(x) is a sequence i.e. what I am calling f(x) is actually $\sum_{n=0}^{x}a_n$ I don't know if this makes a big difference or not.)

Can someone point in the right direction please?

Thanks
Just post the problem itself.

4. There is no particular formula. I just want to know in general if there is some procedure by which I could find out whether

$\frac{\sum_{n=0}^{x}a_n}{(x+1)^3}$

diverges or converges (has a limit).

According to the calculus tutorial thread by TPH, for a regular old sequence $\sum_{n=0}^{x}a_n$ there are various tools to figure out whether the sequence is divergent or convergent (has a limit). The ratio test or divergence theorem for example. Is there some way to apply these tools to the fraction above?

(aside: why does Latex mess up my sigma notation when I put it into a fraction?)

5. Originally Posted by tonio
This is a weird question: we have to know what $f(x)$ or, at least, its behavior when $x\rightarrow \infty$ to have some chance to answer..

Tonio

perhaps im being a knucklehead, but since the following is true
$
$
$\lim_{x\to \infty }\frac{\sum_{n=0}^{x}a_n}{(x+1)^3}\leq\lim_{x\to \infty }\frac{Cx}{(x+1)^3}$ where $C =\max(|a_n|) \;for\; n=0,...,x$, aren't we done. The expression is convergent to 0.

6. Originally Posted by vince
perhaps im being a knucklehead, but since the following is true
$\lim_{x\to \infty }\frac{\sum\limits_{n=0}^x a_n}{(x+1)^3}\leq\lim_{x\to \infty }\frac{Cx}{(x+1)^3}$ where $C =\max(|a_n|) \;for\; n=0,...,x$, aren't we done. The expression is convergent to 0.

I wouldn't say a "knucklehead" , but definitely wrong, imo: since C depends on x, we can't have what you wrote...

Now , if $\{a_n\}$ is a(n overall) bounded sequence then yes, we could write what you did.

Tonio

7. Originally Posted by tonio
I wouldn't say a "knucklehead" , but definitely wrong, imo: since C depends on x, we can't have what you wrote...

Now , if $\{a_n\}$ is a(n overall) bounded sequence then yes, we could write what you did.

Tonio

nice one...indeed.