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Thread: Limit problem: lim [n -> infty] [(1 + 2^p + 3^p + ... + n^p)/(n^{p+1})]

  1. #1

    Limit problem: lim [n -> infty] [(1 + 2^p + 3^p + ... + n^p)/(n^{p+1})]

    Hello

    I was wondering which areas this math problem involves in? I'm currently reading a course in calculus and I do not recognize the problem in my course literature .
    Would be grateful if you could list the parts that you need to solve this problem.

    . . .[tex]\mbox{For }\, p\, >\, 0,\, \mbox{ solve the following:}[/tex]

    . . . . .[tex]\displaystyle \lim_{n \rightarrow \infty}\, \dfrac{1\, +\, 2^p\, +\, 3^p\, +\, ...\, +\, n^p}{n^{p+1}}[/tex]
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  2. #2
    Elite Member stapel's Avatar
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    Cool

    Quote Originally Posted by emmablom View Post
    I was wondering which areas this math problem involves in? I'm currently reading a course in calculus and I do not recognize the problem in my course literature .
    Would be grateful if you could list the parts that you need to solve this problem.

    . . .[tex]\mbox{For }\, p\, >\, 0,\, \mbox{ solve the following:}[/tex]

    . . . . .[tex]\displaystyle \lim_{n \rightarrow \infty}\, \dfrac{1\, +\, 2^p\, +\, 3^p\, +\, ...\, +\, n^p}{n^{p+1}}[/tex]
    What was the source of this exercise? If your textbook, then you should probably apply topics from that section, and immediately previous.

    Note: Wolfram Alpha gives the limit as being zero, if 0 < p < 1. (here) So I'm guessing that you're probably expected to consider cases....

  3. #3

    part chapter ?

    The question is from a previous test in the course and I wonder what part chapter I can read in order to fully understand the problem?


    This is what my course literature covers.

    Iterations

    Newton-Raphson's Method

    Series: convergence / divergence, integral criterion

    comparison criteria

    root / quota criteria

    Alternating series

    Leibniz criterion

    rearrangement of series

    Power series

    Taylor series

    Power series convergence

    Derivation / integration of power series

    solution of differential equations

    Sequences, series of functions

    pointwise and uniform convergence

    Dominated convergence

    reversal of border processes.

  4. #4
    Elite Member stapel's Avatar
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    Cool

    Quote Originally Posted by emmablom View Post
    The question is from a previous test in the course and I wonder what part chapter I can read in order to fully understand the problem?

    . . .[tex]\mbox{For }\, p\, >\, 0,\, \mbox{ solve the following:}[/tex]

    . . . . .[tex]\displaystyle \lim_{n \rightarrow \infty}\, \dfrac{1\, +\, 2^p\, +\, 3^p\, +\, ...\, +\, n^p}{n^{p+1}}[/tex]

    This is what my course literature covers.

    Iterations

    Newton-Raphson's Method

    Series: convergence / divergence, integral criterion
    . . . . ..comparison criteria
    . . . . ..root / quota criteria
    . . . . ..Alternating series
    . . . . ..Leibniz criterion
    . . . . ..rearrangement of series
    . . . . ..Power series
    . . . . ..Taylor series
    . . . . ..Power series convergence
    . . . . ..Derivation / integration of power series

    solution of differential equations

    Sequences, series of functions

    pointwise and uniform convergence

    Dominated convergence

    reversal of border processes
    I would guess that the sequences-and-series chapter(s) would be a good place to start. But you know your book better than I do. You're taking the course; where does this sort of exercise arise in the book's homework sets? Study that section, and the preceding ones.

  5. #5
    thanks for the tip!

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