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 Apr13 revised Proof by induction valid or not? added 1169 characters in body Apr13 answered Proof by induction valid or not? Apr13 comment For any continuous function f(x), how can I split up the function and restrict the domain to find an inverse? This is a pretty broad question. You might be better off coming up with one or a few simple examples of problems you want to be able to solve and trying to work through those, maybe with help from this community if you get stuck. Apr13 answered basic conditional probability proof Apr12 answered Prove $a_n = \begin{cases}1 & \text{if$n=2^k$for some$k\in N$}\\ 0 & \text{otherwise} \end{cases}$ diverges. Apr12 comment When are $\Delta x$, $\delta x$, $dx$, and $\text{đ}x$ exactly the same? When are they approximately the same? when you write $\delta x$, do you mean $\partial x$? Apr12 awarded Citizen Patrol Apr12 revised Discrete Math: Inductions latex Apr12 reviewed Approve Householder matrix Uw acts as the identity on the subspace w Apr12 reviewed Approve Can I calculate this sum using matrix multiplication? Apr11 comment Why do some series converge and others diverge? Let's for the moment restrict our attention to series with nonnegative decreasing terms. I think about it kind of like this. There are two opposing "forces" pushing on the behaviour of the partial sums of this series. On the one hand, each of the partial sums must be at least as big as as the last one, since we are adding more and more terms. This pushes up on the partial sums. On the other hand, with each term, the increases in the partial sums are decreasing, since the terms are getting smaller. If the series converges, it is because this second force is somehow stronger than the first. Mar13 answered Number of words in containing $0,1$ Mar13 comment Modus operandi for proving Evaluation Fundamental Theorem of Calculus (Abbott p 200, Spivak p 272 T14.2) But this clearly does not necessarily hold for all $x\in (t_i - t_{i-1})$'s, so we need to show that there is an $x_i$ in each interval such that it holds. But that is exactly what the mean value theorem promises. Mar13 comment Modus operandi for proving Evaluation Fundamental Theorem of Calculus (Abbott p 200, Spivak p 272 T14.2) @TuckerRapu $f$ can be discontinuous. MVT is a very powerful theorem that you can use to prove all kinds of things about the behaviour of a differentiable function on an interval. As littleO explains, we can get to $g(b)-g(a)\approx \sum g'(x_i)(t_i - t_{i-1})$ for some $x_i$ in the interval without appealing the the mean value theorem. Now we want to tighten that $\approx$ to an $=$, that is, we want to write $g(b)-g(a) = \sum g'(x_i)(t_i - t_{i-1})$... Mar13 answered GCD of pairs of integers Mar13 revised A difficult equation containing exponent 2 and 3 edited tags Mar13 answered A difficult equation containing exponent 2 and 3 Mar13 answered Composite Relations - Give Examples of relations $R_1$ and $R_2$ such that $R_2 \circ R_1 = R_1 \circ R_2$ and $R_2 \circ R_1 \neq R_1 \circ R_2$ Mar13 comment Composite Relations - Give Examples of relations $R_1$ and $R_2$ such that $R_2 \circ R_1 = R_1 \circ R_2$ and $R_2 \circ R_1 \neq R_1 \circ R_2$ I suspect that in definition of a relation you meant to write $$R_2 \circ R_1 = \{(x,y) \in S \times S :( \exists \mathbf{v} \in S)[(x,v) \in R_1 \land (v,y) \in R_2]\}$$ rather than $\exists y$. Mar13 answered Modus operandi for proving Evaluation Fundamental Theorem of Calculus (Abbott p 200, Spivak p 272 T14.2)