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reviewed Approve How to understand $E(XY)$ intuitively
Jul
23
reviewed Approve How to draw graphs of functions?
Jul
23
comment How to draw graphs of functions?
gnuplot, free online services like SageMathCloud, SymPy Gamma
Jul
23
answered Coefficients of general Fourier Series
Jul
19
reviewed Approve INDUCTION: Let a sequence of numbers $a_n$ for $n\in \mathbb N$ be defined by the following rule: $a_1 = 1$, and for $n>1$, $a_n = 2a_{n-1} + 1$
Jul
19
reviewed Approve How to compute an expected value in shorter ways (when taking all possibilities into account isn't plausible.)
Jul
17
comment What is the solution(s) to $1^i$?
$e^{ix}$ is an $2\pi$-periodic function, thus we have $(e^{2kπi})^i=1^i, \ k\in\mathbb{Z}.$
Jul
14
reviewed Approve Prove the value of this infinite series
Jul
10
reviewed Approve integral question help me please
Jul
8
reviewed Approve Product of all elements in finite group
Jul
8
comment Is this multivariable function continuous?
What can you say about continuity at $(x,\,y)=(1,\,1)?$
Jul
5
reviewed Reject Generating vectors in a non-orthogonal 3D lattice with increasing magnitude
Jul
3
reviewed Approve $L^{\infty}$ norm
Jul
3
reviewed Approve Integral of rational function with a squared term in the denominator
Jul
3
reviewed Approve Coin distribution problem to optimize
Jul
2
comment suppose $a_n>1$ $a_n$is non-decreasing and bounded. prove: $\sum_{n=-1}^{\infty}(1-\frac{a_n}{a_{n+1}})\frac{1}{\sqrt{a_{n+1}}}$ converges
First note that the sequence $\left\lbrace{a_{n}} \right\rbrace$ converges since it is non-decreasing and bounded: $\exists \lim\limits_{n\to\infty}{a_n}=a.$ Thus the sequence of partial sums converges too: $$S_N=\sum\limits_{n=1}^{N-1}(a_{n + 1} - a_n)=a_2-a_1+a_3-a_2+\ldots+a_N-a_{N-1}=a_{N}-a_1\underset{n\to\infty}{\to}a-a_1‌​$$ The second point is that the sequence $\left\lbrace\frac{1}{a_{n + 1}^{3/2}} \right\rbrace$ is monotone and bounded, so we can apply the Abel's test.
Jun
29
reviewed Approve Two prove two lines in a triangle are parallel
Jun
28
reviewed Approve Classify relations “is greater than or equal to”
Jun
26
reviewed Approve Probability of selecting same numbers by players (not a pigeon hole problem)
Jun
25
answered Two girl's ages are $20$ when added and $99$ when multiplied. What is the age of each girl?