Represent sinh2x as power series

In summary, the first three non zero terms of the power series representation of f(x) = sinh 2x are 2x + 8x^3/3! + 32x^5/5!. It is also possible to use the general formula for a Taylor series or express sinh(x) in terms of exponentials to find the power series.
  • #1
synergix
178
0

Homework Statement



find the first three non zero terms of a power series representation of f(x)= sinh 2x

Homework Equations



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The Attempt at a Solution



seems easy enough do I just substitute 2x for x?

so sinh 2x= 2x + 8x3/3! + 32x5/5!
 
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  • #2
Yes that should be correct.
 
  • #3
Thanks I messed this one up on a test because I forgot to put the formula on my cheat sheet but damn that was easy
 
  • #4
You didn't really need to have the formula on your cheat sheet if you know how to differentiate sinh(2x). Just use the general formula for a Taylor series.
 
  • #5
Dick said:
You didn't really need to have the formula on your cheat sheet if you know how to differentiate sinh(2x). Just use the general formula for a Taylor series.

Or if you know how to express sinh(x) in terms of exponentials, and you know the Taylor series for exp(x).
 
  • #6
Well I tried using Taylor series and am happy to say that was also very easy and much more satisfying. Those Taylor series aren't so bad after all:)
 

1. What is the power series representation of sinh2x?

The power series representation of sinh2x is 2x + (4/3)x^3 + (8/15)x^5 + (16/105)x^7 + ...

2. How is this power series derived?

The power series representation of sinh2x is derived using the Taylor series expansion for the hyperbolic sine function. This involves taking derivatives of the function at x = 0 and substituting them into the general formula for a Taylor series.

3. How accurate is this power series representation?

The accuracy of the power series representation of sinh2x depends on the number of terms used in the series. The more terms that are included, the closer the approximation will be to the actual value of the function.

4. Can this power series be used to approximate values for any value of x?

Yes, the power series representation of sinh2x can be used to approximate values for any real number x. However, the convergence of the series may be slow for large values of x.

5. Are there any other ways to represent sinh2x?

Yes, there are other ways to represent sinh2x, such as using the Maclaurin series or the Euler's formula for the hyperbolic sine function. However, the power series representation is often preferred because it is easier to manipulate and can be used for a wider range of values for x.

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