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Use Euler's method with step size $10^{-n}$ for $n=1,2,3,4.$ to estimate $x(1)$, where $f(x)$ is the solution of the initial-value problem below.

$x'=f(x)=-x$

$x(0)=1$

EDIT / UPDATE:

x_n+1=x_n + f(x_n)h

how does it become x_n+1= x_n[(1-h)^(n+1)]

Solution then is X(1)=X_n where n=[1/h]

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    thanks, Khanak; reposting/editing keeps the proble$m$/work altogether, in one location, and when you edit/answer a problem, it gets kicked up to the front of the "active" posts, so it won't get lost or overlooked. I'll add "EDIT" to your post, so those who have read the post previously see your update more readily....2012-10-31

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Euler's method is such:

$ \begin{align*} x_0 &= x(0), \\ x_n &= x_{n-1}+hf(x_{n-1}). \end{align*}$

For $h = 10^{-n}$, $x_{10^n} = x(1)$.