# Tagged Questions

Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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### O Notation and taylor series

Wolframalpha tells me that the Taylor series of the exponential function is $1 + x + \frac{x^2}{2}+ O(x^3).$ Taylor series I just don't get this big O there, shouldn't this be a small o?
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### additive subgroup of real numbers with non empty interior

G is an additive subgroup of real numbers with a nonempty interior.Then G is all the real numbers.what is the exact proof?
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### Showing that if the $n$th derivative of a function is bounded then it is real analytic

I reproduce from my lecture notes: Suppose $f$ is $C^\infty$ on $[a,b]$ with $$\left|f^{(n)}(x)\right|\leqslant M~~\text{for all}~~x\in(a,b).$$ Then $f$ is real analytic in $[a,b]$. Proof. ...
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### If $\{f_n\}$ and $\{g_n\}$ be uniformly convergent sequences of bounded functions on S, then $\{f_ng_n\}$ is uniformly convergent on S.

If $\{f_n\}$ converges uniformly to $f$ and $\{g_n\}$ converges uniformly to $g$, does it mean $\{f_ng_n\}$ will converge uniformly to $fg$? I am absolutely stuck on this. Please help.
I'm reading Courant's Introduction to Calculus and Analysis. In the introduction, he shows some examples of limits of sequences, the sequence in question is: $$a_n=\frac{n^2-1}{n^2+n+1}$$ Then he ...
Given the D'Alembert operator D'Alembertian $\Box$, I want to show that $$G(x,t,x_0,t_0):= \frac{\delta \left(t_0 + \frac{||x-x_0||}{c} -t \right)}{||x-x_0||}$$ satisfies  \Box G(x,t,x_0,t_0) = ...