Final answer:
The radius of convergence, R, of the series ∑[infinity]ₙ₌₀(-1)ⁿ - (x)²ⁿ/(2n) can be determined using the ratio test. The radius of convergence is √2.
Step-by-step explanation:
The radius of convergence, R, of the series ∑[infinity]ₙ₌₀(-1)ⁿ - (x)²ⁿ/(2n) can be determined using the ratio test. The ratio test states that if ∑[infinity]ₙ₌₀aₙ converges, then the ratio of consecutive terms, |aₙ₊₁/aₙ|, approaches a limit L as n approaches infinity. To find the radius of convergence, we need to find the values of x for which the limit of |(-1)ⁿ - (x)²ⁿ/(2n₊₁) / ((-1)ⁿ - (x)²ⁿ/(2n))| approaches zero.
Let's apply the ratio test to the given series:
|(-1)ⁿ₊₁ - (x)²ⁿ₊₁/(2n₊₂)| / |(-1)ⁿ - (x)²ⁿ/(2n)|
We can simplify this expression to:
|(-1)ⁿ * ((-1) - (x)²/(2n₊₂))| / |(-1)ⁿ - (x)²ⁿ/(2n)|
Since |(-1)ⁿ| = 1, the expression simplifies to:
|((-1) - (x)²/(2n₊₂))| / |1 - (x)²/(2n)|
As n approaches infinity, the term containing n in the denominator becomes negligible, and the expression reduces to:
|1 - (x)²/2|
This limit approaches zero when |1 - (x)²/2| < 1, which can be further simplified to |(x)²/2| < 1. Solving this inequality gives us -√2 < x < √2. Therefore, the radius of convergence, R, is √2.