Find the sum of infinite terms of the series: 1 + 2x + 3x2 + 4x3 + ... where |x| < 1.
Explanation:
Let S = 1 + 2x + 3x2 + 4x3 + ... (1)
This is an Arithmetic Geometric Progression (AGP). The coefficients are in AP where as the variable is in GP.
Step 1: Multiply the whole equation with x. ⇒ xS = x + 2x2 + 3x3 + 4x4 + ... (2)
Now, (1) - (2) ⇒ (1 - x)S = 1 + (2x - x) + (3x2 - 2x2) + (4x3 - 3x3) + ... ⇒ (1 - x)S = 1 + x + x2 + x3 + ...
Now, RHS is an infinite GP whose first term is 1 and common ratio is x. ∴ (1 - x)S = 1/(1 - x) ⇒ S = 1/(1 - x)2
Hence, option (b).
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