Question

Use a Maclaurin series in this table to obtain the Maclaurin series for the given function. f(x) = x cos(4x)

Answer 1

The Maclaurin series for the given function f(x) = x cos(4x) is f(x) = 1 - 16x².

Step-by-step explanation:

The Maclaurin series expansion for a function involves expressing the function as an infinite sum of its derivatives evaluated at a specific point (usually 0). In this case, we can use the Maclaurin series for cos(x) and multiply it by x to obtain the series for x cos(4x).

The Maclaurin series for cos(x) is
`\( \cos(x) = 1 - (x^2)/(2!) + (x^4)/(4!) - (x^6)/(6!) + \ldots \)`.

Multiplying this by x gives
`\( x \cos(x) = x - (x^3)/(2!) + (x^5)/(4!) - (x^7)/(6!) + \ldots \)`.

Now, substitute 4x for x to obtain the series for xcos(4x):

`\[ f(x) = 4x - ((4x)^3)/(2!) + ((4x)^5)/(4!) - ((4x)^7)/(6!) + \ldots \]`

Simplify each term and combine like terms:

`\[ f(x) = 4x - (64x^3)/(2) + (1024x^5)/(24) - (16384x^7)/(720) + \ldots \]`

Further simplification gives the final answer f(x) = 1 - 16x². This is obtained by keeping only the terms up to x² and combining them. Therefore, the Maclaurin series for f(x) = xcos(4x) is f(x) = 1 - 16x².

Answer 2

The Maclaurin series for the function
`\( f(x) = x \cos(4x) \)` is given by
`\( \sum_(n=0)^(\infty) (-1)^n (2^(2n)x^(2n+1))/((2n+1)!) \)`.

The Maclaurin series is obtained by expressing the given function as an infinite sum of its derivatives evaluated at the origin. For
`\( f(x) = x \cos(4x)`\), we use the Maclaurin series expansion for
`\( \cos(4x) \)`, which is
`\( \sum_(n=0)^(\infty) (-1)^n ((4x)^(2n))/((2n)!) \)`. Multiplying this by
`\( x \)`, we get
`\( \sum_(n=0)^(\infty) (-1)^n (2^(2n)x^(2n+1))/((2n+1)!) \)`.

In this series, each term corresponds to the nth derivative of
`\( f(x) \)`evaluated at
`\( x = 0 \)`. The general term follows the pattern of the nth derivative of
`\( \cos(4x) \)`multiplied by
`\( x \)`. The alternating signs come from the expansion of
`\( \cos(4x) \)`. This series converges for all real numbers, providing an approximation of the original function
`\( f(x) \)` in the vicinity of the origin. The accuracy of the approximation increases as more terms are included in the series.