Question

Use the equation 1x = { x for [x< 1 to expand the function (Use symbolic notation and fractions where needed.) n=0 6 Σ 6(x+6)" in+1 1 X n=0 Incorrect Determine the interval of convergence

Answer 1

The question asks to expand a function using the binomial theorem and determine the interval of convergence. The expansion is done by applying the formula for the binomial theorem, while the interval of convergence can be found using tests like the Ratio Test or the Root Test.

Step-by-step explanation:

The question seems to refer to expanding a power series and finding its interval of convergence. While the original equation provided in the question is unclear due to typographical errors, the concept of working with series and determining their interval of convergence is a key topic in calculus, particularly in the study of infinite series. The binomial theorem and properties of convergence are often used to expand functions and determine the range over which the series is valid.

In practice, to expand a function using the binomial theorem, we use the formula ^n = a^n + n*a^n-1*b + /2!*a^n-2*b^2 + ... and so on. The interval of convergence for a power series can be found using various tests such as the Ratio Test or the Root Test, which provide conditions that determine whether the infinite series converges or diverges.

Answer 2

The interval of convergence for the given series is
`\((-7, 1)\).`

Explanation:

The interval of convergence for a power series is determined by finding the values of \(x\) for which the series converges. In this case we are given the series
`\(\sum_(n=0)^(\infty) (6(x+6)^n)/(n+1)\)`. To determine the interval of convergence, we use the ratio test, which states that a series
`\(\sum_(n=0)^(\infty) a_n\)`converges if
`\(\lim_(n\to\infty) \left|(a_(n+1))/(a_n)\right| < 1\)`. Applying this to our series we get:

`\[\lim_(n\to\infty) \left|((6(x+6)^(n+1))/(n+2))/((6(x+6)^n)/(n+1))\right| < 1\]`

`Simplifying, we find \(\lim_(n\to\infty) (n+1)/(n+2) |x+6| < 1\). Solving for \(x\), we get \(-7 < x+6 < 1\), which yields the interval \((-7, 1)\).`

In summary, by applying the ratio test to the given series, we found that it converges within the interval
`\((-7, 1)\)`. This means the series converges for all x values within this range. The steps involved in the ratio test and the algebraic manipulations were crucial in arriving at this conclusion.