Question

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Find the Maclaurin series for the function. (Use the table of power series for elementary functions.)
f(x)=eˣ⁸ / 8
f(x)= ₙ₌₀∑[infinity] ___

Answer 1

The Maclaurin series for the function f(x) = e¹⁸ / 8 is obtained by substituting x⁸ for x in the standard e¹ series and then dividing the entire series by 8. The final series is ∑ ₙ=₀ [∞ (x⁸ⁿ/n!)/8], summed from n = 0 to infinity.

Step-by-step explanation:

To find the Maclaurin series for the function f(x) = eˣ⁸ / 8, we start by considering the Maclaurin series for e˚, which is a well-known expansion:

e˚ = 1 + x + x²/2! + x³/3! + ... + xⁿ/n! + ...

Now, for f(x) = eˣ⁸ / 8, we can substitute x⁸ for x in the series of e˚ and then divide the entire series by 8:

f(x) = (1 + (x⁸) + (x⁸)²/2! + (x⁸)³/3! + ... + (x⁸)ⁿ/n! + ...)/8

This gives us the Maclaurin series for f(x):

f(x) = ∑ ₙ=₀ [∞ (x⁸ⁿ/n!)/8]

Where n! denotes factorial of n, and the series is summed over n from 0 to infinity.