Chapter 2: Q22P (page 71)

Using (12.2) and (8.1), find, in summation form, the power series for sinhxand coshx. Check the first few terms of your series by computer.

Short Answer

Expert verified

$The Power of \sinh (x) = [1 + \frac{x^{3}}{3!} + \frac{x^{5}}{5!} + . . .] And \cosh (x) = [1 + \frac{x^{2}}{2!} + \frac{x^{4}}{4!} + . . .] .$

Step by step solution

Step 1: Given Information.

The given expression is sinh (x) , cosh (x) ..

Definition of Power Series.

A Power Series which is in one variable is an infinite series written in the form of $\sum_{n = 0}^{\infty} a_{n} {(x - c)}^{n} = a_{0} + a_{1} (x - c) + a_{2} {(x - c)}^{2} + . . . Where a_{n}$

Whererepresents the coefficient of the term and represents the constant.

Formula’s to be used in solution.

$Let ’ s state the notation to be used in the solution . \cosh (ix) = \cosh (iz) \sinh (ix) = \sinh (iz) e^{z} = {\sum \frac{z^{n}}{n!}}_{0}^{\infty} \sinh (z) = \frac{e^{z} - e^{- z}}{2} \cosh (z) = \frac{e^{z} - e^{- z}}{2}$

Find the Power series of and.

$\sin h (z) = \frac{1}{2} [e^{z} - e^{- z}] = [\sum_{0}^{\infty} \frac{z^{n}}{n!} - \sum_{0}^{\infty} \frac{{(- z)}^{n}}{n!}] = [\sum_{0}^{\infty} \frac{z^{n}}{n!} - \sum_{0}^{\infty} \frac{{(- 1)}^{n} z^{n}}{n!}]$

If n is odd then the notation has a value, but if n is even then the notation is zero.

$\sin h (z) = \frac{1}{2} [\sum_{0}^{\infty} \frac{z^{n}}{n!} - \sum_{0}^{\infty} \frac{{(- 1)}^{n} z^{n}}{n!}] = \frac{1}{2} [(1 - 1) + (z + z) (\frac{z^{2}}{2!} - \frac{z^{2}}{2!}) + (\frac{z^{3}}{3!} - \frac{z^{3}}{3!}) + . . .] = \frac{1}{2} [2 z + 2 x \frac{z^{3}}{3!} + 2 \times \frac{z^{5}}{5!} + . . .] = [z + \frac{z^{3}}{3!} + \frac{z^{5}}{5!} + . . .] \cos h (z) = \frac{1}{2} [e^{z} + e^{- z}] = \frac{1}{2} [\sum_{0}^{\infty} \frac{z^{n}}{n!} + \sum_{0}^{\infty} \frac{{(- z)}^{n}}{n!}] = \frac{1}{2} [\sum_{0}^{\infty} \frac{z^{n}}{n!} + \sum_{0}^{\infty} \frac{{(- z)}^{n}}{n!}]$

Hence,

The Power of

$\sinh (x) = [x + \frac{x^{3}}{3!} + \frac{x^{5}}{5!} + . . .] And \sinh (x) = [1 + \frac{x^{2}}{2!} + \frac{x^{4}}{4!} + . . .] .$

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Using (12.2) and (8.1), find, in summation form, the power series for sinhxand coshx. Check the first few terms of your series by computer.

Short Answer

Step by step solution

Step 1: Given Information.

Definition of Power Series.

Formula’s to be used in solution.

Find the Power series of and.

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