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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)

^{n}a

^{n}f(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)

^{n}a

^{n}f(x) = (-1)

^{0}a

^{0}f(x) + (-1)af(x) + (-1)

^{2}a

^{2}f(x) + (-1)

^{3}a

^{3}f(x)

= f(x) - af(x) + a

^{2}f(x) - a

^{3}f(x).

For a --- is this correct? I thought the series itself was (-1)

^{n}a

^{n}f(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