# How do you find the Maclaurin series for f(x) using the definition of a Maclaurin series, of 4 sinh(4x)?

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We derive a Maclaurin series from the infinite series

We have:

Differentiate wrt to get the first derivative:

Differentiate again wrt to get the second derivative:

Differentiate again wrt to get the third derivative:

Differentiate again wrt to get the fourth derivative:

Differentiate again wrt to get the fifth derivative:

And we can see a clear pattern forming, where

So we can now form the Maclaurin series:

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To find the Maclaurin series for ( f(x) = 4\sinh(4x) ), we first need to determine the Maclaurin series expansion for ( \sinh(x) ). The Maclaurin series for ( \sinh(x) ) is:

[ \sinh(x) = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \frac{x^7}{7!} + \cdots ]

Since we have ( 4\sinh(4x) ), we substitute ( 4x ) for ( x ) in the series expansion of ( \sinh(x) ):

[ 4\sinh(4x) = 4(4x) + \frac{(4x)^3}{3!} + \frac{(4x)^5}{5!} + \frac{(4x)^7}{7!} + \cdots ]

Simplify this expression to obtain the Maclaurin series for ( f(x) = 4\sinh(4x) ):

[ f(x) = 16x + \frac{64x^3}{3!} + \frac{256x^5}{5!} + \frac{1024x^7}{7!} + \cdots ]

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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