Let(1+x)^2(1+(x)/(2))^2(1+(x)/(2^2))^2(1+(x)/(2^3))^2 ⋯upto ∞=a0+a1 x+a2 x2+⋯upto ∞, then (a) a1

Let(1+x)^2(1+(x)/(2))^2(1+(x)/(2^2))^2(1+(x)/(2^3))^2 ⋯upto...

Key Concepts

Infinite Product Representation

This concept concerns representing a function as a product of an infinite number of factors. In analysis, infinite products can serve to define functions or express them in a way that reveals structure not obvious in other forms. Handling such products requires careful convergence considerations and an understanding of how operations like expansion interact with the infinite nature of the product.

Power Series Expansion and Coefficient Extraction

Expanding a function as a power series involves writing it as an infinite sum of terms, each multiplied by a power of the variable. Coefficient extraction is the process of determining the coefficients corresponding to each power. This technique is essential in solving problems that ask for specific term values in the series, often by identifying contributions from multiple sources within a product or sum representation.

Geometric Series

A geometric series is a series where each term is obtained by multiplying the previous one by a fixed constant. Recognizing a geometric series is crucial because it allows for the summation of an infinite number of terms to a closed-form expression, provided the series converges. This concept is frequently applied in power series problems where sequences of terms arise naturally during expansion.

Transcript

00:01 Hi, it is given that look 1 plus x whole square 1 plus x over 2 whole square 1 plus x over 2 square whole square and so on up to infinity is given as this polynomial up to infinity.

00:12 We're going to find out a1, a2, a3.

00:16 So how are we going to approach this problem here? what you can do here in the given expression 1 plus x whole square 1 plus x over 2 we can replace x with x over 2.

00:29 So we get here 1 plus x over 2 whole square times 1 plus x over 2 square whole square times 1 plus x over 2 cube up to infinity that equals we have a note plus a 1 1 x over x over 2 whole square and so on up to infinity.

01:03 Therefore, we find out here this a notch plus a1x plus a2x squared and so on.

01:09 So we get here a0 plus a1x plus a2x square plus a3 xq and so on to infinity.

01:22 It is given as equal to.

01:25 Now it's given as equal to 1 plus x squared times this.

01:28 Now this part here, we find out that is equal to this equal to this.

01:32 This a node plus a 1 x over 2 and so on so we write it as 1 plus x square 1 plus x whole square times a node plus a 1 x over 2 plus a 2 x over 2 whole square and so on next we're to equate here the coefficient of x to the power r on the left hand side and right hand side.

02:05 So what are doing now, we're now equating coefficient of x to the power r.

02:18 So for that, what we can do, we can just take here, for example, let's say x square on both sides.

02:23 So i'll take that.

02:25 Maybe we have x square and on the right hand side we have for this term here that's coming out to be x squared.

02:35 So when i equate the coefficient of x to power r, so i will get a r equal, and the first thing is here we have a 2 x square.

02:46 And the right hand side we have here x squared.

02:49 So first thing is we have x square will multiply with then we have one.

03:02 One will multiply with this part x squared.

03:05 Again we get the coefficient of x squared and this part here x over 2.

03:08 It will multiply with 2x here from this.

03:12 That's how we're using this to minus 4 x to power r.

03:15 The first term is coming out to be, we get for ar, that is coming out to be 1 over 2 to the power r, a.

03:24 That term we got from where? so that term we got from here.

03:28 1 over 2 to the part 2, here we have a2.

03:31 So we take a general term, that is 1 over 2 to the power r, a.

03:35 Next time it's coming out to be plus two times 1 over 2 to the part r negative 1, a, r negative 1...

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