8.. Compute the power series centered at zero for f(x) = coshx in two different ways. a. Find the power series by using the standard approach for computing a Maclaurin series. Show all the derivatives and evaluation of derivatives clearly. Use the facts that cosh'(x) = sinh(x), sinh'(x)=cosh(x), cosh(0)=1 and sinh(0) = 0 rather than expanding the hyperbolic functions in terms of exponentials. b. Find the power series by using the standard power series for e*. and coshx=-(e² +e™²). Write the exponentials in the cosh function as series expansions, then clearly show the algebra as you simplify the sum of the two series. You should arrive at the same power series from part n! n=0 a. Recall that e* =

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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8.. Compute the power series centered at zero for f(x) = coshx in two different ways.
a. Find the power series by using the standard approach for computing a Maclaurin series.
Show all the derivatives and evaluation of derivatives clearly. Use the facts that
cosh'(x) = sinh(x), sinh'(x) = cosh(x), cosh(0) = 1 and sinh(0) = 0 rather than expanding the hyperbolic functions in terms
of exponentials.
b. Find the power series by using the standard power series for e*.
I'm!
=1¹(e* + e^²). Write the exponentials in the cosh function as series expansions, then
n=0
clearly show the algebra as you simplify the sum of the two series. You should arrive at the same power series from part
a.
Recall that e* =
and cosh.x=-
Transcribed Image Text:8.. Compute the power series centered at zero for f(x) = coshx in two different ways. a. Find the power series by using the standard approach for computing a Maclaurin series. Show all the derivatives and evaluation of derivatives clearly. Use the facts that cosh'(x) = sinh(x), sinh'(x) = cosh(x), cosh(0) = 1 and sinh(0) = 0 rather than expanding the hyperbolic functions in terms of exponentials. b. Find the power series by using the standard power series for e*. I'm! =1¹(e* + e^²). Write the exponentials in the cosh function as series expansions, then n=0 clearly show the algebra as you simplify the sum of the two series. You should arrive at the same power series from part a. Recall that e* = and cosh.x=-
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