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 Yearling
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Apr
27
comment $\int^{2 \pi}_0 \frac{1}{3+2 \cos t}dt$ using $\cos t = \frac{1}{2}\left(e^{it} + \frac{1}{e^{it}}\right)$ or using $u=\tan \frac{t}{2}$
Note that $u$ blows up at $t=\pi$, and is not a bijection on $[0,2\pi]$, so you will have to be careful about how you make that substitution.
Apr
14
comment Proving $\langle f,f \rangle =0 \implies f=0$
@snowman By your definition, $F(1)=\int_{0}^{1} |f(x)|^{2} dx = 0$. We do not know anything (yet) about $\int_{0}^{t}|f(x)|^{2} dx$ for other values of $t$.
Apr
14
comment Proving $\langle f,f \rangle =0 \implies f=0$
But you haven't proved it's an inner product yet! Think about it this way: suppose you're trying to convince someone who doesn't believe you that $<f,g>$ is an inner product or that $<f,f>$ is a norm. Try to prove the statement without assuming either of those facts.
Apr
14
comment Proving $\langle f,f \rangle =0 \implies f=0$
Well, if you do assume it then the fact is trivial. But someone, somewhere has to prove it! As we note above, the condition only tells you that $F(1)=0$, not that $F(t)=0$ for all $t$.
Apr
14
comment Proving $\langle f,f \rangle =0 \implies f=0$
@snowman The condition only tells you that $F(1)=0$.
Apr
14
answered Proving $\langle f,f \rangle =0 \implies f=0$
Apr
12
comment Why does the same equation have different results?
I would word it the following way: Let $x$ be the decimal expansion $0.999...$ Then $10x-x=9x$ as real numbers. But the decimal expansion represented by $10x-x$ is $9.000...$, while the decimal expansion represented by $9x$ is $8.999...$. While these decimal expansions are different, they must represent the same real number.
Apr
12
answered Why does the same equation have different results?
Apr
12
comment Why does the same equation have different results?
I would also argue that it's kind of obvious that was the intention, but hey ho.
Apr
1
answered Continuity and differentiability problem and checking is the function elelment of $C^0$ and $C^1$
Mar
29
comment Real matrices with $n$ real positive eigenvalues, $A^2=B^2$. Prove that $A=B$
The question was recently edited to include that.
Mar
29
comment Real matrices with $n$ real positive eigenvalues, $A^2=B^2$. Prove that $A=B$
Are the eigenvalues distinct?
Mar
29
answered Pluralisation when describing multiple objects simultaenously
Mar
22
answered How to actually prove that the chracterstic polynomial is monic and of degree n
Mar
20
comment Prove that $ \lim_{x \to \infty} f^{n}(x) = 0$
Nope, the condition for L'hopital is that the top and bottom tend to zero (or, equivalently, infinity). That might not be true.
Mar
20
comment Prove that $ \lim_{x \to \infty} f^{n}(x) = 0$
That doesn't prove anything - L'Hopital's rule doesn't even apply here, since you don't know $f(x) \to 0$.
Mar
18
awarded  Yearling
Mar
15
comment uniformly continuous functions on infinite open interval $(0, \infty)$
Moreover, its derivative is $1/x^{2} (x\cos(x)-\sin(x))$ which is bounded.
Mar
9
answered Test of Convergent $\sum_{n = 1}^{\infty} \left| \log{\left( n \sin{\frac{1}{n}} \right)} \right|$
Mar
1
answered Limit of a sequence inside a set