Recent content by TheMathSorcerer

It's probably worth making an attempt using the first FTC and understanding it. You could use this, Set ##F(x) = \int_{\alpha(x)}^{\beta(x)}{ f(t) dt}##. Then, ##F'(x) = f(\beta(x))\beta'(x) - f(\alpha(x))\alpha'(x)## However it's probably worth understanding the first FTC before jumping...

You messed up, it's definitely worth fixing and understanding. You should get ##\lim n \to \infty \frac{x}{n + 1} = 0##. Convince yourself:) Here ##L = 0##. The ratio test says the series converges absolutely if ##L < 1##. For what values of ##x## is this true?

The order need not be ##xy##. Take ##Z_2## which is cyclic with generator ##1##. Then in the group ##Z_2 \times Z_2##, the element ##(1,1)## has order two since ##(1, 1) \neq (0,0)##, and ##(1, 1) + (1, 1) = (0, 0)##, yet ##Z_2 \times Z_2## has order ##4##, so ##(1,1)## does not generate ##Z_2...