Infinite sum

Calculus Level pending

∑ j = 0 ∞ ( 3 j + 2 ) 3 x j + 1 \large \sum_{j=0}^\infty (3j+2)^3 x^{j+1}

If the series above converges and can be expressed as x ( x 3 − a x 2 + b x + c ) ( x − 1 ) 4 \dfrac{x(x^3 - ax^2 + bx + c)}{(x-1)^4} for a constant x x . Find the value of a a .

1 -205 -60 -19 93

Akira Kato by Akira Kato , 22, China

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1 solution

Akira Kato
Oct 19, 2016

Upon expansion, the sum is S = 2 3 a + 5 3 a 2 + 7 3 a 3 + 9 3 a 4 . . . S=2^{3}a+5^{3}a^{2}+7^{3}a^{3}+9^{3}a^{4}... . Through algebraic manipulation, it becomes S ( 1 − a ) 3 = 8 a + 101 a 2 + 161 a 3 + 162 ( a 4 / ( 1 − a ) S(1-a)^{3}=8a+101a^{2}+161a^{3}+162(a^{4}/(1-a) thus S = 8 a + 93 a 2 + 60 a 3 + a 4 = 0 S=8a+93a^{2}+60a^{3}+a^{4}=0 Apply vieta's formula for sum of all root, you get -60/1=-60

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