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1d
comment An exercise from Knuth's book - Proving a formula by induction
Your second approach does work, you have to simplify the difference, and you'll find $0$. Regarding the knuth tag, notice he's an author, not a mathematical field by himself. It would not be wise to create a tag for each good math textbook.
1d
comment Approximation by polynomials
If $P(x)$ is a good approximation of $f$, then $P(x^4)$ is a good approximation of $f(x^4)$. Just write what a good approximation means...
1d
comment Does $\exp(2ir\pi)$ equal $1$? What's wrong?
What is wrong here, is that you did not define $a^b$ for a complex number $a$ and a noninteger $b$.
1d
comment Approximation by polynomials
Also, the constant term is not really a burden: if $|P(x)-f(x)|<\varepsilon$ for all $x\in[0,1]$, then $|P(0)-f(0)|<\varepsilon$ and with $P_0=P-P(0)$, you have $|P_0(x)-f(x)|\leq |P_0(x)-P(x)|+|P(x)-f(x)|<2\varepsilon$. Just halve $\varepsilon$.
1d
comment Approximation by polynomials
Regarding your second question, notice that $P=Q(x^{1/4})$ is then an arbitrary polynomial. Hence the question is: given a continuous function $g$ such that $g(x)=f(x^{1/4})$, is it possible to find a polynomial $P$ such that $|g(x)-P(x)|<\varepsilon$? Weierstrass' theorem answers this question. It won't work as is for negative $x$, but notice your $f$ is even, and so is the polynomial $Q(x)=P(x^4)$.
1d
comment Approximation by polynomials
Regarding "I thought that this is a corollary of thhe approxmiation Theorem, but I think one has to prove that $a_0$ has to be zero": If you are asked to prove that for each $\varepsilon>0$ there is a polynomial $P$, then Weierstrass' theorem is enough. If you have to prove that the is a $P$ that works for each $\varepsilon>0$, it's wrong anyway, unless $f$ is a polynomial on $[0,1]$ (just make $\varepsilon\to0$, then for any $x$, $P(x)-f(x)\to0$, so $P(x)=f(x)$).
1d
comment Taylor Formula: Lagrange's remainder vs Cauchy's remainder (and other less known forms)
@Dal Indeed :-)
1d
comment Taylor Formula: Lagrange's remainder vs Cauchy's remainder (and other less known forms)
It does not answer the question, but you may also be interested by the Schlömilch remainder. Here is a note about the different remainders and their derivation (in french). The french Wikipedia article about Taylor's theorem has also some information about these (and also the so-called Taylor-Young and Taylor-Laplace).
1d
comment If every proper subgroup of a group is finite, does it follow that the group is finite?
mathoverflow.net/questions/120829/…
1d
comment Reverse an equation with ln and power
Please don't add "thank you" as an answer. Once you have sufficient reputation, you will be able to vote up questions and answers that you found helpful.
Dec
13
comment How do we pronounce this symbol?
@IncnisMrsi This makes sense. In physics, though, we always spelled it "nabla phi", not "del phi", but it may be a french usage. Thank you anyway.
Dec
12
comment The product of two nonnegative, improperly integrable functions is also improperly integrable.
What about a function with rectangular pikes at $x=n$ of area $1/n$ (width $1/n$, height $1$), and alternating sign? The limit of the integral would exist, but $f^2$ is not integrable, since it's the same as $|f|$, and $f$ is not absolutely integrable.
Dec
12
comment Finding the inverse of a recursive function?
@SalmonKiller Given that recursive functions are all computable functions, you may imagine there is a whole lot of complicated cases. Are you asking for a general procedure for ivnersion? There is simply no such thing. So basically yes, you have to find some pattern.
Dec
12
comment How to prove a double sum is always an integer?
Not necessarily useful, but notice that $\dfrac{1}{s+1}{2s\choose s}$ can be put in factor, and this is a Catalan number. Similar remark for your 2nd equality.
Dec
11
comment Algorithm to numerically solve this system of three polynomial equations of degree $6$
There are some references here on Wikipedia, for example have look at PHCPack (see here and algo 795 at TOMS Collected Algorithms. If you have an access, you may also have a look at recent publications (ACM and SIAM come to mind).
Dec
10
comment Proof of mean for a log normal distribution
Yes, it's as simple as this: substitution $y=x-\mu$. The integral does not change if you integrate on $]-\infty,+\infty[$ (notice you didn't write the bounds). But here you integrate the pdf of a normal distribution, this has nothing to do with the mean of a log-normal distribution, so why this title?
Dec
10
comment Find the function of integer numbers $\sum_{n=0}^{\infty }\frac{n^k}{n!}=f(k) \cdot e$
A bit wrong as you state it, but this is Dobinski's formula. The integers are Bell numbers. The proof on Wikipedia, using factorial moments of Poisson distribution, is a classic.
Dec
10
comment What does the “or” symbol mean as in “$ d\mid a$”
@Talespin_Kit en.wikipedia.org/wiki/Divisor
Dec
10
comment What does the “or” symbol mean as in “$ d\mid a$”
@Talespin_Kit Yes, that's the meaning of "divides" here.
Dec
9
comment What is the meaning of |⋯| notation for an index subset?
No, $|I|=10$ and $|J|=5$, and $I=\{1,2,3,4,5,6,7,8,9,10\}$, $J=\{1,2,3,4,5\}$.