Question

Find the point of intersection of the following two lines

r₁ (t)=⟨−4,2,2⟩+t⟨−9,3,4⟩
r₂ (t)=⟨12,0,−4⟩+s⟨10,−5,−5⟩
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Answer 1

To represent the function f(x) = 8x²(x+4) as a power series centered at 0, you can use a geometric series. The power series representation is 8x² + 32 + (32/x) + (32/x)^2 + (32/x)^3 + ... The radius of convergence is 0.

Step-by-step explanation:

To represent the function f(x) = 8x²(x+4) as a power series centered at 0, we can use a geometric series. We start by factoring out the common factor of 8x² from the equation: f(x) = 8x²(x+4) becomes f(x) = (8x²)(1 + 4/x). Now we can express the second term, 1 + 4/x, as a geometric series: 1 + 4/x = 1 + (4/x) + (4/x)² + (4/x)³ + .... Finally, we substitute this geometric series back into the original equation to get the power series representation: f(x) = (8x²)(1 + 4/x) = 8x² + 32 + (32/x) + (32/x)² + (32/x)³ + ....

To determine the radius of convergence of the power series, we can use the formula for the radius of convergence of a power series. In this case, the series involves terms of the form (32/x)^n, so we need to find the values of x that make this expression converge. The radius of convergence is the distance from the center of the series (0 in this case) to the nearest singularity of the function represented by the series. In this case, the function f(x) has a singularity at x = 0, so the radius of convergence is 0.