Infinite Series familiar function

[carlodelmundo]

Homework Statement

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

i. f(0) = 1

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

The Attempt at a Solution

a.) Here is my work for:
$$\sum$$$$^{\infty}_{n=0}$$ (-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

 

[Mark44]

Hint: e^kx is a familiar function

 

[HallsofIvy]

Homework Statement

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

i. f(0) = 1

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

The Attempt at a Solution

a.) Here is my work for:
$$\sum$$$$^{\infty}_{n=0}$$ (-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