Consider the initial value problem $$\dfrac{dy}{dx} = y,y(0) =1$$ Approximate $y(1)$ using Euler's method with a step size of $\dfrac{1}{n}$, where $n$ is an arbitrary natural number. Use this approximation to write Euler's number $e$ as a limit of an expression in $n$. How large do you have to choose $n$ in order to approximate $e$ up to an error of at most 0.1? Comment on the quality of approximate in this example.
What I did is the following:
$y(1) \approx y_1 = y_0 +hf(x_0,y_0) = 1+hf(0,1) = 1+\dfrac{1}{n}$
This is where I stuck, am I on the right direction? What should I do next?
So, you have found $y_1$ using $x_0$ and $y_0$. To find $x_1$, recall that $x_1 = x_0 + h = \frac{1}{n}$ (in general, $x_k = x_0 + k h = x_{k - 1} + h$). Once you find $x_1$, you can find $y_2$ by using $$ y_2 = y_1 + h f(x_1, y_1) = \left(1 + \frac{1}{n}\right) + \frac{1}{n} f\left( \frac{1}{n}, 1 + \frac{1}{n} \right) = 1 + \frac{2}{n} + \frac{1}{n^2} = \left( 1 + \frac{1}{n} \right)^2. $$ Can you see that pattern above? What do you think $y_k$ will be? (HINT: $y_3 = \left( 1 + \frac{1}{n} \right)^3$ ) Keep repeating. That is, find $x_n$, then $y_{n+1} = y_n + h f(x_n, y_n)$. That will help you express $e$ as a limit.