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By observing that f(x) = 1/(1 - 2x) is the sum of a geometric series (of the form a/(1 - r)), find the power series expansion of this function. Observation of Series: We'll observe the value of the first term or numerator aa and the common ratio r (it is the quotient of the second term to the first term) in the denominator of the rational function a1−ra1−r, plug these values in the formula of expansion. a1−r=a+ar+ar2+ar3+…a1−r=a+ar+ar2+ar3+… Where |r||r| is less than one.

Answer :

The power series of the given function  is 1 + (2x) + (2x)² + (2x)³+ --------------------- + (2x)ⁿ

We know very well that sum of n terms who are in geometric progression their sum of expression is given by  a/1-r where a is first term and r is common ratio between the terms.

Now, we have function f(x)=[1 / (1-2x)]

On comparing with a/1-r with f(x),we get

=>a=1 and r=2x

Now, we know that first term of geometric progression is given by =1

second term of geometric progression is given by=a × r= 1 ×2x

third term of geometric progression is given by =a×r² =1×(2x)²

fourth term of geometric progression is given by=a×r³ =1 × (2x)³

nth term of geometric progression is given by =a×(r)ⁿ = 1 × (2x)ⁿ

Therefore, according to the given formula progression series of given function is=a+ ar +ar² + ar³ + -----------arⁿ

=>progression series = 1+ 2x + (2x)² + (2x)³ + --------- + (2x)ⁿ.

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