Determine a power series for f(x)=6/(1+〖7x〗^4 ) and give the interval of convergence
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Because it wanted to expand its knowledge!
To find the power series representation of the function f(x) = 6/(1 + 7x^4), we can use the geometric series formula:
1/(1 - u) = ∑ (u^n)
where |u| < 1.
In this case, u = -7x^4, so we need |u| < 1 to ensure convergence:
|-7x^4| < 1
7x^4 < 1
x^4 < 1/7
Taking the fourth root of both sides gives us:
x < (1/7)^(1/4)
So the interval of convergence is (-((1/7)^(1/4)), ((1/7)^(1/4))).
Now, let's find the power series representation:
f(x) = 6/(1 + 7x^4)
= 6 * (1/(1 + 7x^4))
Substituting u = -7x^4, we have:
= 6 * ∑ (u^n)
= 6 * ∑ (-7x^4)^n
= 6 * ∑ (-7)^n * (x^4)^n
= 6 * ∑ (-7)^n * (x^4)^n
= 6 * ∑ (-1)^n * 7^n * x^(4n)
Therefore, the power series representation for f(x) is:
f(x) = 6 * ∑ (-1)^n * 7^n * x^(4n)
where the interval of convergence is (-((1/7)^(1/4)), ((1/7)^(1/4))).
To determine a power series representation for the function f(x) = 6/(1+7x^4), we can use the geometric series formula.
The geometric series formula states that for a series of the form 1/(1-r), the sum of the series can be expressed as a power series when |r| < 1.
In this case, our function f(x) can be written as 6 * (1/(1+7x^4)). We can rewrite 1/(1+7x^4) as a geometric series by noticing that the ratio between consecutive terms is (7x^4).
Now, we will substitute (7x^4) into the geometric series formula.
The formula for the sum of a geometric series is given by:
S = a/(1-r)
where
S is the sum of the series,
a is the first term of the series, and
r is the ratio between consecutive terms.
In this case, the first term a is 6 and the ratio r is 7x^4. Substituting these values into the formula, we get:
S = 6/(1-7x^4)
Simplifying further, we have:
S = 6 * (1 + 7x^4 + (7x^4)^2 + (7x^4)^3 + ...)
This is a power series representation of the function f(x), which converges for values of x such that |7x^4| < 1.
To find the interval of convergence, solve the inequality:
|7x^4| < 1
Divide both sides by 7:
|x^4| < 1/7
Take the fourth root of both sides:
|x|^(4/4) < (1/7)^(1/4)
Simplifying, we have:
|x| < (1/7)^(1/4)
So, the interval of convergence for the power series representation of f(x) is (-[(1/7)^(1/4)], (1/7)^(1/4)).