Question

Using Maclaurin series, determine to exactly what value the series converges.

∑ from n=0 to [infinity] (-1)ⁿ * ((4π)²ⁿ / (2ⁿ)!)

Answer 1

The series converges to the value of the cosine of 4π, which is equivalent to cos(0), and thus the series converges to exactly 1.

Step-by-step explanation:

The question involves determining the convergent value of a series represented by a Maclaurin series. The series in question is ∑ from n=0 to ∞ (-1)ⁿ * ((4π)²ⁿ / (2ⁿ)!). This series is actually the expansion for the cosine function, given by the Maclaurin series cos(x) = ∑ from n=0 to ∞ (-1)ⁿ * (x²ⁿ / (2n)!). In our case, x is equal to 4π, and so the series converges to cos(4π). Since the cosine function has a period of 2π, cos(4π) is equivalent to cos(0), which is exactly 1.