So do that, and see where it leads.My only thinking at the moment is to bring theto the front and try to bring in that rule to solve for
 \lim\limits_{n \to \infty} \sum_{i=1} ^n \ cos {\frac{k \pi}{4n}} )
Having done further work on it last night, I dont feel like Im getting closer to the answer. Try to used the trig identity to simplify but am coming up with uncancellable terms.
After reading what you said, my thinking is the sin limit can get reduced to 1?
The limit you have written in this post is not true, the numerator tends to 0 and the denominator is a nonzero constant, so the fraction you have written converges to 0.After reading what you said, my thinking is the sin limit can get reduced to 1?
}{sin(\frac{\pi}{8})}=1 )
Let the antiderivative of
Not sure how to go about that one, thinking Fundamental Theorem of Calculus is involved.
My thinking :
Not really he could just set f'(x)=I think you need to replace your f(x) with F(x), with F'(x) = f(x)=.
The FTC applicable in this case (the first FTC according to wikipedia, although nomenclature varies) tells you that if f(t) is a continuous function on [a,b], thenNot really he could just set f'(x)=and solve the DE
It is just ambiguous.
And yes he can use the FTC, but does he need to check or at least affirm the conditions. f has to be C1? (or if using your notation F)
on the interval of question.
Is this inequality basically an application of (continuous version of) Jensen's inequality?It is good practice to try to prove (weighted) AM-GM in this setting.
Suppose f,w are non-negative functions on [0,1] that are continuous (or more generally just Riemann-integrable) and suppose that w has integral 1 on this interval.
Prove that:
Try to carefully justify each step of your argument as much as possible.
Sure, that is one way to do it. (In exactly the same way the discrete weighted Jensen's proves discrete weighted AM-GM). I would expect in such an answer that a student would prove the continuous form of Jensen's though, as the main emphasis of the exercise is the generalisation of discrete inequalities to continuous inequalities rather than the proof of the AM-GM inequality itself.
what do I need to knowIt is good practice to try to prove (weighted) AM-GM in this setting.
Suppose f,w are non-negative functions on [0,1] that are continuous (or more generally just Riemann-integrable) and suppose that w has integral 1 on this interval.
Prove that:
Try to carefully justify each step of your argument as much as possible.
Pretty much just the definition of Riemann integrability (See https://en.wikipedia.org/wiki/Riemann_integral) and basic properties of limits. It is not a difficult question in terms of technicality.
Here's a cool one to think about:
_{k=0}^\infty$ of real numbers, does there exist a a smooth function $f\in\mathcal{C}^\infty(\mathbb{R})$ such that\\ \\ $\frac{d^kf}{dx^k}(0)=a_k$ for each $k$?$\\ \\ )
Here is an exercise to exhibit a couple of quirks of high dimensional solids. (Not as directly related to first year calculus courses as most things in this thread, but it is first year level stuff, and it is in the calculus/analysis ballpark.)
a) By generalising MX2 methods of volume calculation, find an expression for the volume of the n-dimensional ball of radius r in terms of the Gamma function
b) How does this quantity behave asymptotically as n->inf? Interpret this as a comparative statement about n-dimensional balls and n-dimensional cubes.
c) What happens to |B(r-d)|/|B(r)| as n-> inf? Here B(r) denotes the n-dimensional ball of radius r, and d < r is fixed. Interpret this result as a statement about the asymptotic concentration of mass in high dimensional balls.
d) Show that the the limiting behaviour in (c) can occur even if d(n) depends on n and monotonically decreases to zero. For which power rates of decay d(n)=n^(-p) will this happen?
e) Repeat c) and d), this time for strips about the equator. That is, what can we say about the limiting behaviour of |{x in B(r): |x_n| < d}|/|B(r)|? What does this say about the asymptotic concentration of mass in high dimensional balls? Can we take d(n)->0 and still have the same behaviour? At what power rate can d tend to zero with us having the same limiting behaviour?
Related:
^n$ and $\lim_{n\rightarrow \infty }\left(\sum_{k=0}^n \frac{x^k}{k!}\right)$ both exist and coincide for all real numbers $x$. $)
^ Bunch of unanswered questions of varying difficulty.Here is a question related to a basic tool in the analysis I do.
-\sum_{j=1}^N b_j \lambda^{-\rho_j})$ is bounded for each positive integer $N$. Think of Taylor series but exponents in such an expansion need not be integers, and the formal series $\sum_{j=1}^\infty b_j \lambda^{-\rho_j}$ need not converge to be a valid asymptotic expansion. $)
When you originally had equality put up I was flummoxed.It is good practice to try to prove (weighted) AM-GM in this setting.
Suppose f,w are non-negative functions on [0,1] that are continuous (or more generally just Riemann-integrable) and suppose that w has integral 1 on this interval.
Prove that:
Try to carefully justify each step of your argument as much as possible.