All Questions
26
questions
16
votes
2
answers
1k
views
Is $\lim\limits_{k \to \infty}\left[ \lim\limits_{p \to \infty} \frac{M}{1+3+5+\cdots+ [2^{p(k-1)}-2^{p(k-2)}-2^{p(k-3)}-\cdots-1]}\right]=1$?
Firstly, my $\LaTeX$, Mathematics and English knowledge is very limited. It is extremely difficult for me to ask this question. Now I am improving myself.I hope you understand me...
Look at this ...
- 375
3
votes
2
answers
2k
views
N'th derivative with chain rule
I am trying to find a general form of the chain rule for higher derivatives, using the general Leibniz rule I got to the following formula. However it doesn't seem to work. I suspect it has something ...
- 436
3
votes
1
answer
88
views
If $\sum_{n=1}^{\infty} \sum_{k=1}^{\infty} f_{n}(k) = \infty$ then $\sum_{k=1}^{\infty} \sum_{n=1}^{\infty} f_{n}(k) = \infty$.
Suppose $\{f_{n}\}_{n=1}^{\infty}$ be functions such that $f_{n} : \Bbb{N} \rightarrow \Bbb{R}^{+}$ for each $n$.
I was trying to prove -
If $\sum_{n=1}^{\infty} \sum_{k=1}^{\infty} f_{n}(k) = \...
- 5,042
2
votes
2
answers
77
views
How to compute taylor series $f(x)=\frac{1}{1-x}$ about $a=3$?
How to compute taylor series $f(x)=\frac{1}{1-x}$ about $a=3$? It should be associated with the geometric series. Setting $t=x-3,\ x=t+3$, then I don't know how to continue, could someone clarify the ...
- 229
2
votes
1
answer
140
views
Partial fractions for any function?
I recently discovered a property of polynomials that have all roots distinct. That is, if $a_i \neq a_j$ for $i \neq j$, and if
$$f(x) = \prod_{i=1}^{n} (x-a_i)$$
Then,
$$\sum_{i=1}^{n} \frac {f(x)}{(...
- 363
2
votes
1
answer
136
views
Sum of the inverse of all positive integers which do not contain the digit 8 [duplicate]
Let $No8$ be the set of positive integers that do not contain the digit $8$. For example, $123456790 ∈ No8$ but $1234567890 \notin No8$. Show that $$\sum_{n\in No8} \frac 1n<80$$
The bound in the ...
user388085
2
votes
2
answers
91
views
How can I prove the concavity of $f(p_1,p_2,\ldots,p_n) = \sum_{i = 1}^n p_i(1-p_i)$?
Assume $p_n$ is the probability of being in class $n$
which mean that $f(0) = 0$ , $f(1) =0$ , and $p_1+p_2 = 1$
I need to come up with a concave function that show the relation between $p_1$ and $...
- 21
2
votes
0
answers
67
views
Closed form for $\psi^{1/k}(1)$, where $k$ is an integer
I have proven the identity
$$
\sum_{k=1}^{\infty} \dfrac{\operatorname{_2F_1}(1, 2, 2-1/t,-1/k)}{{k}^{2}} = Γ(2-\dfrac{1}t){\psi^{1/t}(1)}+\psi(-\dfrac{1}t)(\dfrac{1}t(1-\dfrac{1}t))+\gamma(1-\dfrac{1}...
2
votes
0
answers
93
views
N'th derivative of $f(x)$ to the power of $g(x)$
I am trying to find a formula for the $n$'th derivative of $f(x)^{g(x)}$. I have tried using the formula bellow along with Leibniz rule without success.
$$D^n(f(g(x)))=\sum_{k=0}^{n-1} \binom {n-1}{k}...
- 436
1
vote
4
answers
957
views
Summation of n-squared, cubed, etc. [duplicate]
How do you in general derive a formula for summation of n-squared, n-cubed, etc...? Clear explanation with reference would be great.
- 493
1
vote
3
answers
548
views
Find the area of the region $y=2x$, the $x$-axis, lines $x=1$ and $x=4$
Find the area of the region $y=2x$, the $x$-axis, lines $x=1$ and $x=4$
Here's what I did:
$$A = \lim_{n \to +\infty} \sum^{n}_{i=1}f(x_{i-1})\Delta x
\\ = \lim_{n \to +\infty} \sum^{n}_{i=1} 2(i-1)\...
- 1,249
1
vote
1
answer
77
views
If $|f(p+q)-f(q)|\leq p/q$ for rational $p$ and $q$ (with $q\neq 0$), then $\sum_{i=0}^k |f(2^k) -f(2^i)| \leq k(k-1)/2$
If $\left|f(p+q) - f(q)\right|\leq p/q$ for all rational $p$ and $q$, with $q \neq 0$, then prove that
$$\sum_{i=0}^k \left|f\left(2^k\right) -f\left(2^i\right)\right| \leq \frac12 k(k-1)$$
My try:...
- 800
1
vote
1
answer
93
views
is f(x) increasing or decreasing? on $(-1, 0]$
Let the function $f(x)$ be defined by $f(x)= \sum_{n=0}^{\infty} \frac{(-1)^{n-1}}{(2n+1)!}x^n$. Since f(x) is a function, then it can increase and decrease. So, in the interval $(-1, 0]$, is f(x) ...
- 119
1
vote
0
answers
57
views
How to approximate $\prod\limits_{(i, j)\in I^2} (1 - \alpha_i x_{i, j})\;$ as a linear function?
Suppose we have a finite set $I \subseteq \mathbb{N}$, and $\alpha_i\in [0, 1]$ are fixed numbers for all $i\in I$.
Is there a way to approximate $\prod\limits_{(i, j)\in I^2} (1 - \alpha_i x_{i, j})...
- 342
1
vote
0
answers
62
views
What is the the function $f(x)$ has the property $\sum_{n=1}^x f(x^n)=x^2$
What is the the function $f(z)$ has the property $\sum_{n=0}^z f(z^n)=z^2$ when $z$ is a natural number bigger than $0$, and the function $f(z)$ is differentiable everywhere.
$$f(1)+f(1)=1$$
$$f(1)+f(...
user913631