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Math Help - does this diverge or converge?

  1. #1
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    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
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  2. #2
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    Quote Originally Posted by rainer View Post
    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
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  3. #3
    Ted
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    Quote Originally Posted by rainer View Post
    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.
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  4. #4
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    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?)
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  5. #5
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    Quote Originally Posted by tonio View Post
    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
    <br />
\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.
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  6. #6
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    Quote Originally Posted by vince View Post
    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
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  7. #7
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    Quote Originally Posted by tonio View Post
    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.
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