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Infinite Series familiar function

1. Homework Statement

Let f(x) be a function with the following properties.

i. f(0) = 1

ii. For all integers n [tex]\geq[/tex] 0, the 0, the nth derivative, f (n)(x) = (-1)nanf(x), where a > 0 and a [tex]\neq[/tex] 1.

a.) Write the first four non-zero terms of the power series of f(x) centered at zero, in terms of a.

b.) Write f(x) as a familiar function in terms of a.

c.) How many terms of the power series are necessary to approximate f(0.2) with an error less than 0.001 with a = 2?

3. The Attempt at a Solution

a.) Here is my work for:
[tex]\sum[/tex][tex]^{\infty}_{n=0}[/tex] (-1)nanf(x) = (-1)0a0f(x) + (-1)af(x) + (-1)2a2f(x) + (-1)3a3f(x)

= f(x) - af(x) + a2f(x) - a3f(x).

For a --- is this correct? I thought the series itself was (-1)nanf(x) given from the above but I think it may be wrong?

for b.)... what does it mean write f(x) as a familiar functions in terms of a? I don't get how we can be "familiar" with this function... I'm guessing it has to do something with the initial condition, f(0) = 1? but what? hints please.

for c.) c is contingent on my answer on b... but I do know we're using the LaGrange Remainder.

Thanks for the help
 
32,573
4,303
Hint: ekx is a familiar function
 

HallsofIvy

Science Advisor
Homework Helper
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1. Homework Statement

Let f(x) be a function with the following properties.

i. f(0) = 1

ii. For all integers n [tex]\geq[/tex] 0, the 0, the nth derivative, f (n)(x) = (-1)nanf(x), where a > 0 and a [tex]\neq[/tex] 1.

a.) Write the first four non-zero terms of the power series of f(x) centered at zero, in terms of a.

b.) Write f(x) as a familiar function in terms of a.

c.) How many terms of the power series are necessary to approximate f(0.2) with an error less than 0.001 with a = 2?

3. The Attempt at a Solution

a.) Here is my work for:
[tex]\sum[/tex][tex]^{\infty}_{n=0}[/tex] (-1)nanf(x) = (-1)0a0f(x) + (-1)af(x) + (-1)2a2f(x) + (-1)3a3f(x)

= f(x) - af(x) + a2f(x) - a3f(x).
That's not even a power series! The power series for f(x), about 0 (the MacLaurin series) is f(0)+ f'(0)x+ f"(0)/2! x^2+ ...+ f(n)(0)/n! x^n+ ...
It should be easy to determine the derivatives of f evaluated at 0.

For a --- is this correct? I thought the series itself was (-1)nanf(x) given from the above but I think it may be wrong?

for b.)... what does it mean write f(x) as a familiar functions in terms of a? I don't get how we can be "familiar" with this function... I'm guessing it has to do something with the initial condition, f(0) = 1? but what? hints please.

for c.) c is contingent on my answer on b... but I do know we're using the LaGrange Remainder.

Thanks for the help
 

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