What is the Maclaurin Series for #tanax#?

Answer 1

# tan ax = ax + 1/3a^3x^3 +2/15a^5x^5 + ...#

The Maclaurin series is given by

# f(x) = f(0) + (f'(0))/(1!)x + (f''(0))/(2!)x^2 + (f'''(0))/(3!)x^3 + ... (f^((n))(0))/(n!)x^n + ...#

We start with the function

# f^((0))(x) = f(x) = tanax #

Then, we compute the first few derivatives:

# f^((1))(x) = (sec^2ax(a) # # \ \ \ \ \ \ \ \ \ \ \ \ = a sec^2(ax) #
# f^((2))(x) = (2a sec^2ax)(secax tanax)(a) # # \ \ \ \ \ \ \ \ \ \ \ \ = 2a^2 sec^2ax tanax # # \ \ \ \ \ \ \ \ \ \ \ \ = 2a^2 (1+tan^2ax) tanax # # \ \ \ \ \ \ \ \ \ \ \ \ = 2a^2 (tanax+tan^3ax) #
# f^((3))(x) = 2a^2{asec^2ax+3atan^2ax sec^2ax} # # \ \ \ \ \ \ \ \ \ \ \ \ = 2a^3sec^2ax{1+3tan^2ax} # # \ \ \ \ \ \ \ \ \ \ \ \ = 2a^3sec^2ax{1+3(sec^2ax-1)} # # \ \ \ \ \ \ \ \ \ \ \ \ = 2a^3sec^2ax{1+3sec^2ax-3} # # \ \ \ \ \ \ \ \ \ \ \ \ = 6a^3sec^4ax-4a^3sec^2ax #
# vdots #
Now we have the derivatives, we can compute their values when #x=0#
# f^((0))(x) = 0 # # f^((1))(x) = a # # f^((2))(x) = 0 # # f^((3))(x) = 2a^3 # # vdots #

Which permits us to form the Maclaurin serie:

# f(x) = (0) + (a)/(1)x + (0)/(2)x^2 + (2a^3)/(6)x^3 + ... (f^((n))(0))/(n!)x^n + ...#
# \ \ \ \ \ \ \ = ax + 1/3a^3x^3 + 2/15a^5x^5 + ... #
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Answer 2

The Maclaurin series for ( \tan(ax) ) is:

[ \tan(ax) = ax + \frac{a^3 x^3}{3} + \frac{2a^5 x^5}{15} + \frac{17a^7 x^7}{315} + \cdots ]

This series expansion is valid for values of ( x ) close to zero.

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Answer from HIX Tutor

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.

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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