Can I Derive the Taylor Series and Radius of Convergence for Tanh(x)?

In summary, The conversation discusses how to derive the Taylor series expansion and radius of convergence for hyperbolic tangent tanh(x) around the point x=0, with a specific focus on finding the expression for the nth order derivative of tanh(x) in terms of Bernoulli numbers. The radius of convergence is determined to be the same as it is for tangent, pi/2. However, the question still remains on how to derive the nth coefficient of the Taylor series and the radius of convergence using methods like the root test.
  • #1
kercalc
2
0
Hi.

How can I derive the Taylor series expansion and the radius of convergence for hyperbolic tangent tanh(x) around the point x=0.

I can find the expression for the above in various sites, but the proof is'nt discussed. I guess the above question reduces to how can I get the expression for the n^th order derivative of tanh(x) in terms of Bernoulli numbers.

Many thanks in advance.
joel.
 
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  • #2
The radius of convergence for tanh is the same as it is for tan, which is pi/2.
 
  • #3
Thanks. But my questions are (a) How can I derive (not just the expression) the n^{th} coefficient of the Taylor series for tanh(x) around x=0, and (b) the derivation of the radius of convergence (by e.g. root test).
 

1. What is the formula for the Taylor series representation of tanh(x)?

The formula for the Taylor series representation of tanh(x) is
tanh(x) = x - (1/3)x^3 + (2/15)x^5 - (17/315)x^7 + (62/2835)x^9 + ...

2. How many terms are needed to accurately approximate tanh(x) using the Taylor series?

The number of terms needed to accurately approximate tanh(x) using the Taylor series depends on the desired level of accuracy. Generally, using more terms in the series will result in a more accurate approximation.

3. What is the interval of convergence for the Taylor series of tanh(x)?

The interval of convergence for the Taylor series of tanh(x) is -1 < x < 1. This means that the series will converge to the exact value of tanh(x) for any value of x within this interval.

4. Can the Taylor series of tanh(x) be used to approximate values outside of the interval of convergence?

No, the Taylor series of tanh(x) can only be used to approximate values of tanh(x) within the interval of convergence (-1 < x < 1). Attempting to use the series outside of this interval may result in inaccurate approximations.

5. How is the Taylor series for tanh(x) derived?

The Taylor series for tanh(x) is derived by taking the Maclaurin series of the function and then using the formula for the nth derivative of tanh(x) to find the coefficients for each term in the series. This process involves finding the derivatives of tanh(x) and evaluating them at x = 0, which results in the simplified form of the series.

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