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Find a power series representation for the function. (Give your power series representation centered at x=0 .) f(x)=15+x

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Answer:


f(x) = \sum\limits^(\infty)_(n=0) (-(1)^n\cdot x^n)/(5^(n+1))

Explanation:

Given


f(x) = (1)/(5 + x)

Required

The power series centered at
x = 0

We have:


f(x) = (1)/(5 + x)

Factor out 5 from the denominator


f(x) = (1)/(5(1 + (x)/(5)))

Rewrite as:


f(x) = ((1)/(5))/((1 + (x)/(5)))

Further, rewrite as:


f(x) = (1)/(5)(1 + (x)/(5))^(-1)

Expand the bracket


f(x) = (1)/(5)(1 - (x)/(5) + ((x)/(5))^2 - ((x)/(5))^3+..........)

Evaluate all exponents


f(x) = (1)/(5)(1 - (x)/(5) + (x^2)/(25) - (x^3)/(125)+......)

Open brackets


f(x) = (1)/(5) - (x)/(5^2) + (x^2)/(5^3) - (x^3)/(5^4)+......

Notice the pattern as:


f(x) = (1)/(5) - (x)/(5^2) + (x^2)/(5^3) - (x^3)/(5^4)+......\± (x^n)/(5^(n+1))

So, the power series is:


f(x) = \sum\limits^(\infty)_(n=0) (-(1)^n\cdot x^n)/(5^(n+1))

User Nevin Paul
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