STEVIE-G™ asked in 科學及數學數學 · 1 decade ago

Interesting Maths 2008

2 Answers

  • Jacob
    Lv 6
    1 decade ago
    Favorite Answer

    First is to find Σ(k=1 to ∞) k^3*r^k (for |r|<1)

    Since Σ(k=1 to ∞) r^k=r/(1-r) for |r|<1

    diff. both side term-by-term w.r.t. r,

    Σ(k=1 to ∞) kr^(k-1)=1/(1-r)^2

    multiply r both side and we get

    Σ(k=1 to ∞) kr^k=r/(1-r)^2

    repeat the same process, we finally get

    Σ(k=1 to ∞) k^3 r^k==r(1+4r+r^2)/(1-r)^4

    putting r=1/3<1, (the series still converges)

    Σ(k=1 to ∞) k^3/3^k=33/8

    Source(s): ME
  • 1 decade ago


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