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Use summation formulas to rewrite the expression without the summation notation .

sum i=1 ^ n 2i^ 3 -3i n^ 4

Use summation formulas to rewrite the expression without the summation notation . sum-example-1
User Cpiock
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1 Answer

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Factor out 1/n⁴, and distribute the sum over each term:


\displaystyle\sum_(i=1)^n(2i^3-3i)/(n^4)=\frac1{n^4}\sum_(i=1)^n(2i^3-3i)


\displaystyle\sum_(i=1)^n(2i^3-3i)/(n^4)=\frac2{n^4}\sum_(i=1)^ni^3-\frac3{n^4}\sum_(i=1)^ni

Recall the following formulas:


\displaystyle\sum_(i=1)^ni=\frac{n(n+1)}2


\displaystyle\sum_(i=1)^ni^3=\frac{n^2(n+1)^2}4

So we have


\displaystyle\sum_(i=1)^n(2i^3-3i)/(n^4)=(n^2(n+1)^2)/(2n^4)-(3n(n+1))/(2n^4)=\boxed{((n+1)(n^2+n-3))/(2n^3)}

User Spacemoose
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