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

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SUMMARY

The Taylor series expansion for the hyperbolic tangent function, tanh(x), around the point x=0 can be derived using the n-th order derivatives of tanh(x) expressed in terms of Bernoulli numbers. The radius of convergence for the Taylor series of tanh(x) is established as π/2, identical to that of the tangent function. The discussion emphasizes the need for a detailed derivation of the n-th coefficient and the application of the root test for determining the radius of convergence.

PREREQUISITES
  • Understanding of Taylor series and their coefficients
  • Familiarity with hyperbolic functions, specifically tanh(x)
  • Knowledge of Bernoulli numbers and their properties
  • Basic concepts of convergence tests, particularly the root test
NEXT STEPS
  • Study the derivation of Taylor series coefficients for hyperbolic functions
  • Research the properties and applications of Bernoulli numbers
  • Learn about convergence tests, focusing on the root test and ratio test
  • Explore advanced calculus topics related to series expansions and convergence
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Mathematicians, calculus students, and anyone interested in series expansions and convergence analysis, particularly those focusing on hyperbolic functions and their applications.

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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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The radius of convergence for tanh is the same as it is for tan, which is pi/2.
 
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).
 

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