In the example of finding the radius of convergence, in part i, because the exponent of n is -2, wouldn’t the ratio test of the (n+1)st term over the nth term result in z(n^2)/[(n+1)^2] rather than x[(n+1)^2]/(n^2)? I’m not sure if I’m missing something or if this was just a typo. Thanks for taking the time to read this:)

]]>I don’t quite understand what you are saying, but here is a proof that the function when and 0 otherwise is not differentiable at 0. Consider the quantity . When it is 0, but when it is , which tends to infinity. So your statement that both limits are finite is simply false.

A small additional remark is that the limits from above and below have to be equal.

]]>In this case you mean that if h \in \mathbb{R}, then f is diffrentialible in a point x if the limit for h->0^+ or h->0^- is finite, but again in the function above both limits are still finite and again the function is not continuos .

Sorry I meant it was always one of my concern about limit’s definition and for me the exception still remains.

Do not ment to be irritant or boring

You say, “Considering you previous definition the following function is differentiable in zero.” However, that is not true. (Hint: can be negative.)

]]>f(x) = 1 for all x >= 0

f(x) = 0 otherwise

Some weird things were going on with me too. I thought I had carelessly typed when I meant and it turned out that I had typed the right thing and it was coming out wrong — but not nonsensically wrong. It seems to be a very strange WordPress bug. I’ve found a fix for it, which is to insert a few spaces where I would ordinarily not need to.

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