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anonymous

  • one year ago

I do not understand how to do this.

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  1. anonymous
    • one year ago
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    Write limn→∞∑k=1n(2+k∗(5/n))3∗5/n as a definite Integral

  2. tkhunny
    • one year ago
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    I can't understand the notation. \(\lim_{n\rightarrow \infty}ln(2+k(5/n))^{3}\cdot\dfrac{5}{n}\)?

  3. anonymous
    • one year ago
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    yes

  4. anonymous
    • one year ago
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    that is the notation

  5. tkhunny
    • one year ago
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    Not quite. What is the "k=" doing in there?

  6. anonymous
    • one year ago
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    im not sure thats what my problem says though. Think it stands for constant

  7. tkhunny
    • one year ago
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    'k' maybe, but '='?

  8. anonymous
    • one year ago
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    oh im sorry let me rewrite equation real quick

  9. tkhunny
    • one year ago
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    Maybe you can use the [Draw] feature?

  10. anonymous
    • one year ago
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    \[\lim_{n \rightarrow \infty}\sum_{k=1}^{n}(2+k*5/n)^3 *5/n\]

  11. tkhunny
    • one year ago
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    Well, that makes more sense!

  12. tkhunny
    • one year ago
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    Did you just learn, in that moment, just enough LaTeX to write that? Good work.

  13. anonymous
    • one year ago
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    Thank you. Now i am unsure of how to convert that to a definite integral

  14. tkhunny
    • one year ago
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    Let's remember the definition of a Riemann Integral. Roughly, you chop things up into smaller and smaller pieces and never stop doing this. This is EXACTLY the function of 'n' in the given expression. \(n \rightarrow \infty\)

  15. anonymous
    • one year ago
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    right

  16. anonymous
    • one year ago
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    still to not understand how to convert

  17. tkhunny
    • one year ago
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    Try a couple values for n and see how it goes.

  18. anonymous
    • one year ago
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    what to do you mean try n values?

  19. tkhunny
    • one year ago
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    Do n=1, then do n = 2. Look for anything familiar or consistent.

  20. anonymous
    • one year ago
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    5*(7k)^3?

  21. tkhunny
    • one year ago
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    The first thing I noticed was that \(5/n\) is constant for a given value of n. This suggests an equivalent expression: \(\lim_{n\rightarrow\infty}\dfrac{5}{n}\sum_{k=1}^{n}\left[2+\dfrac{5k}{n}\right]^{3}\)

  22. anonymous
    • one year ago
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    yes i agree with that

  23. anonymous
    • one year ago
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    and the two is constant

  24. anonymous
    • one year ago
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    can you just give me in converted? I like working back from the conversion

  25. phi
    • one year ago
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    I think of the 5/n as an increment (e.g. dx ) the k* 5/n ranges from 5/n to 5 for very large n, 5/n is close to 0, so k*5/n represents a variable x going from 0 to 5

  26. anonymous
    • one year ago
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    i do not understand what that means

  27. phi
    • one year ago
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    It helps if you have the "big picture" of how integration is summation of lots of thin triangles.

  28. phi
    • one year ago
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    *rectangles

  29. tkhunny
    • one year ago
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    \(=\lim_{x\rightarrow\infty}\dfrac{5}{n}\sum_{k=1}^{n}f\left(\dfrac{5k}{n}\right)\) \(=\lim_{x\rightarrow\infty}\dfrac{b-a}{n}\sum_{k=1}^{n}f(x^{*})\) \(=\int_{0}^{5}f(x)\;dx = \int_{0}^{5}\left(2+5x\right)^{3}\;dx\)

  30. phi
    • one year ago
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    I think it's just (2+x)^3 inside the integral

  31. anonymous
    • one year ago
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    okay! Thank you

  32. tkhunny
    • one year ago
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    Right. Typo. Thanks.

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