Prove that :
( 1+x+x²+x³+...) ( 1-x +x³ - x³+..... ) = (1+x² +x⁴ +x6+...).
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Answered by
11
LHS = ( 1 + X + X² + X³ + ..... ) ( 1 - X + X² - X³ + ..... )
=> ( 1 - X ) ^-1 ( 1 + X ) ^ -1
=> [ ( 1 - X ) ( 1 + X ) ] ^-1
=> ( 1 - X² ) ^ -1
=> ( 1 + X² + X⁴ + X^6 +........ ) = RHS.
=> ( 1 - X ) ^-1 ( 1 + X ) ^ -1
=> [ ( 1 - X ) ( 1 + X ) ] ^-1
=> ( 1 - X² ) ^ -1
=> ( 1 + X² + X⁴ + X^6 +........ ) = RHS.
Answered by
0
( 1+x+x²+x³ +...... ) (1-x+x²-x³+.....)
= (1-x)-¹ (1+x)-¹ = [(1-x)(1+x)]-¹
= (1-x²)-¹
= (1+x²+x⁴+x6+......) = rhs
= (1-x)-¹ (1+x)-¹ = [(1-x)(1+x)]-¹
= (1-x²)-¹
= (1+x²+x⁴+x6+......) = rhs
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