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  • limits - Prove that $\lim \limits_ {n \to \infty} \frac {x^n} {n!} = 0 . . .
    Consider that $$\frac {2^n} {n!} = \frac {\overbrace {2\times 2\times\cdots \times 2}^ {n\text { factors}}} {1\times 2 \times \cdots \times n} = \frac {2} {1}\times \frac {2} {2}\times \frac {2} {3}\times\cdots \times\frac {2} {n} $$ Every factor except the first two is smaller than $1$, so at each step you are multiplying by smaller and smaller numbers, with the factors going to $0$
  • functional analysis - How to prove the proposition on Leray solutions . . .
    Using Cauchy-Schwarz and Young inequality $$ \int_0^t \langle f,u\rangle ds \le \int_0^t \|f\|_ {V'}\|u\|_V ds \le \left (\int_0^t \|f\|_ {V'}^2ds \right)^ {1 2
  • real analysis - Calculating Bernoulli Numbers from $\sum\limits_ {n=0 . . .
    Note that $$\frac {e^z-1}z=\frac1 z\sum_ {n=1}^\infty\frac1 {n!}z^n=\sum_ {n=1}^\infty\frac1 {n!}z^ {n-1}=\sum_ {n=0}^\infty\frac1 { (n+1)!}z^n$$ and we can use Mertens’ multiplication theorem to get $$1=\left (\sum_ {n=0}^\infty\frac {B_n} {n!}z^n\right)\left (\sum_ {n=0}^\infty\frac1 { (n+1)!}z^n\right)=\sum_ {n=0}^\infty\sum_ {k=0}^n\left (\frac {B_k} {k!}\frac {1} { (n-k+1)!}\right)z^n
  • How to prove the proposition on rotating fluids?
    Without really understanding what you are talking about: i) did you intend to write "rotating" instead of "roating" in the title? ii) I'm rather sure even people who would know what your are talking about would find it helpful if you'd give a definition or at least explanation which quantities you are dealing with E g : what is $\delta_k^\varepsilon$? - you seem to be interested in solutions
  • Is zero positive or negative? - Mathematics Stack Exchange
    So what IS the Holy Bible The Great Standardization Document of All Definitions for Mathematics? Because people are often fighting over different definitions of mathematical entities, 0 being one of such examples (French always start a flamewar when someone says 0 is not positive, because for French, 0 is positive and negative at the same time :P ) Same goes with definitions of angles, or
  • mathematical induction ($ (1+x)^n\ge1+nx+n (n-1)x^2 2$)
    also note it says for $n \ge 2$ so your base case needs to be $n=2$
  • real analysis - Is $\sum_ {n=0}^ {\infty} {\frac {2^ {n}x^ {n}} {n . . .
    if anyone could help me with the following problem : Let $\\sum_{n=0}^{\\infty}{\\frac{2^{n}x^{n}}{n!}}$ , I have to see the pointwise convergence on $(0,1)$, I know that the uniform convergence impl





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