Question

express sin(x⁴) as an alternarting series. while you will start with a mclaurin series, your final answer will not have an x's

Answer 1

To express sin(x⁴) as an alternating series, substitute x⁴ for x in the Maclaurin series for sin(x) and simplify to get an alternating series with terms that include powers of x⁴ with alternating signs.

Step-by-step explanation:

We want to express sin(x⁴) as an alternating series using the Maclaurin series for the sine function. Recall that the Maclaurin series for the sine function is given by:

sin(x) = x - x³ / 3! + x⁵ / 5! - x⁹ / 7! + ... + (-1)²(n-1)/2 x²n-1 / (2n-1)! + ...

For sin(x⁴), we will substitute x⁴ for x in the Maclaurin series:

4. Replace every instance of x with x⁴.
8. Apply the substitution step by step to obtain the new series.

The new series becomes:

sin(x⁴) = x⁴ - (x⁴)³ / 3! + (x⁴)⁵ / 5! - (x⁴)⁹ / 7! + ... + (-1)²(n-1)/2 (x⁴)²n-1 / (2n-1)! + ...

As x⁴ is our only variable and there is no x in the final answer, we can simplify the series by evaluating the coefficients:

sin(x⁴) = x⁴ - x⁼ / 3! + x₀ / 5! - x₄ / 7! + ... + (-1)²(n-1)/2 x³(4n-4) / (2n-1)! + ...

This is an alternating series because the signs alternate between positive and negative for each consecutive term, and it follows the (-1)²(n-1)/2 pattern.