Radius of convergence of a series, and approximating polynomials
I hinted today that there were sometimes issues when you did a polynomial approximation, that if you tried to find the value of a function a long way from the region about which you’re approximating, that sometimes you wouldn’t be able to do it. This is related to an idea called the radius of convergence of a series. In the following we are just plotting polynomials, but you can see that whereas in the polynomial approximation for sin(x) (on the right), as we get more and more terms, we approximate the function better and better far away from the point x=1 (which is the point about which we are approximating the function). However, for the function 
is defined as satisfying![f(\lambda x + (1-\lambda )y)\leq \lambda f(x)+(1-\lambda )f(y) \quad \forall x,y\in \mathbb R,\ \forall \lambda \in [0, 1].](http://s0.wp.com/latex.php?latex=f%28%5Clambda+x+%2B+%281-%5Clambda+%29y%29%5Cleq+%5Clambda+f%28x%29%2B%281-%5Clambda+%29f%28y%29+%5Cquad+%5Cforall+x%2Cy%5Cin+%5Cmathbb+R%2C%5C+%5Cforall+%5Clambda+%5Cin+%5B0%2C+1%5D.&bg=ffffff&fg=000&s=0 "f(\lambda x + (1-\lambda )y)\leq \lambda f(x)+(1-\lambda )f(y) \quad \forall x,y\in \mathbb R,\ \forall \lambda \in [0, 1].")
. An example of a convex function is f(μ)=μ2:
and 
. Once you’ve done that, the other parts of the inequalities should be clear. It should look like the red region in the following plot:
