Question

Let S denote the infinite sum 2 + 5x + 9x2 + 14x3 + 20x4 + …, where | x | < 1, then S equals​

Answer 1

S = 2 + 5x + 9x²+ 14x³ + 20x⁴ + ...

Now, multiply S by x :-

Sx = 2x + 5x² + 9x³+ 14x⁴+ 20x⁵ + ...

Now,

S = 2 + 5x + 9x²+ 14x³ + 20x⁴ + ...

-Sx= - 2x - 5x² - 9x³-14x⁴- 20x⁵ + ...

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S-Sx = 2 + 3x +4x² +5 x³ +6x⁴+ ....

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➜ S (1-x) = 2 + 3x +4x² +5 x³ +6x⁴+ ....

Now, again multiply S(1-x) by x :-

➜ S(1-x)x = 2x + 3x² +4x³ +5 x⁴ +6x⁵+ ....

Now,

S(1-x) = 2 + 3x +4x² +5 x³ +6x⁴+

-S(1-x)x = -2x - 3x²- 4x³ - 5 x⁴-6x⁵+ ....

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S(1-x)-S(1-x)x = 2+x + x² + x³ + x⁴+...

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[We can write S(1-x)-S(1-x)x as S(1-x)(1-x).]

➜ S(1-x)(1-x) = 2+x + x² + x³ + x⁴+...

Now, x + x² +x³ + x⁴... is in infinite G.P with first term and common ratio as x. (Also, | x | < 1 )

→ x + x² +x³ + x⁴... = x/(1-x)

➜ S(1-x)(1-x) = 2+ x/(1-x)

➜ S(1-x)² = ( 2-2x+x)/(1-x)

➜ S(1-x)² = ( 2-x)/(1-x)

➜ S = ( 2-x)/(1-x) × 1/(1-x)²

➜ S = ( 2-x)/(1-x)³