Question

Find a power series for the function, centered at c. f(x)= 8/3x+2

Answer 1

The power series representation for the function
`\(f(x) = (8)/(3x+2)\) centered at \(c\) is \(\sum_(n=0)^(\infty)(-1)^n(4)/(3)(x-c)^n\).`

Step-by-step explanation:

The given function
`\(f(x) = (8)/(3x+2)\) can be expressed as a power series using the geometric series formula. First, factor out the denominator to obtain \((8)/(2(1+(3)/(2)x))\). This allows us to write \((8)/(2)\) as \((4)/(3)\) and rewrite \((1)/(1-(-3x)/(2))\) in the form of a geometric series.`

`Now, the series expansion of \((1)/(1-r)\) is \(\sum_(n=0)^(\infty)r^n\) for \(-1 < r < 1\). In our case, \(r = (-3x)/(2)\). Plug this into the formula, and you get \(\sum_(n=0)^(\infty)(-1)^n(4)/(3)\left((3x)/(2)\right)^n\).`

To center the series at a specific point
`\(c\), replace \(x\) with \((x-c)\) in the series. The final power series representation for \(f(x)\) centered at \(c\) is \(\sum_(n=0)^(\infty)(-1)^n(4)/(3)(x-c)^n\).`This series converges within a certain interval around \(c\) where the original function is defined.