All Questions
26
questions
0
votes
1
answer
61
views
Cool identities/properties involving the Alternating Harmonic Numbers
Using the following analytic continuation for the Alternating Harmonic Numbers ($\bar{H}_x=\sum_{i=1}^x\frac{(-1)^{i+1}}i$): $$\bar{H}_x=\ln2+\cos(\pi x)\left(\psi(x)-\psi\left(\frac x2\right)-\frac1x-...
- 6,549
2
votes
0
answers
67
views
Closed form for $\psi^{1/k}(1)$, where $k$ is an integer
I have proven the identity
$$
\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}...
0
votes
2
answers
160
views
$\lim_{{n \to \infty}} \sum_{{k=1}}^{n} \arctan\left(\frac{1}{k}\right) - \ln n$
$\arctan(x) = \sum_{m=0}^{\infty} \frac{(-1)^m}{2m+1}x^{2m+1}$
\begin{align*}
\arctan\left(\frac{1}{k}\right) &= \sum_{m=0}^{\infty} \frac{(-1)^m}{2m+1}\left(\frac{1}{k^{2m+1}}\right)
&= \frac{...
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
0
votes
1
answer
223
views
prove or disprove that "the sum of infinite even or odd functions is even or odd".
i know that if $f_1(x),f_2(x),...,f_n(x)$ is odd/even functions and $c_1,c_2,...,c_n$ are fixed real numbers then $c_1 f_1(x) + c_2 f_2(x) +... +c_n f_n(x)$ is odd/even.
But how for infinite sum ((sum ...
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.
$$f(1)+f(1)=1$$
$$f(1)+f(...
user913631
1
vote
1
answer
93
views
is f(x) increasing or decreasing? on $(-1, 0]$
Let the function $f(x)$ be defined by $f(x)= \sum_{n=0}^{\infty} \frac{(-1)^{n-1}}{(2n+1)!}x^n$. Since f(x) is a function, then it can increase and decrease. So, in the interval $(-1, 0]$, is f(x) ...
- 119
2
votes
0
answers
93
views
N'th derivative of $f(x)$ to the power of $g(x)$
I am trying to find a formula for the $n$'th derivative of $f(x)^{g(x)}$. I have tried using the formula bellow along with Leibniz rule without success.
$$D^n(f(g(x)))=\sum_{k=0}^{n-1} \binom {n-1}{k}...
- 436
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
0
votes
1
answer
59
views
Integral from infinite sum
I am trying to find if there is a way of turning the following sum into an integral:
$$\lim_{x\to\infty} \sum_{k=1}^{x}\operatorname{tr}\left(\frac{C^{k}}{x^2+x}\right)$$
I have looked into Riemann ...
- 241
0
votes
0
answers
85
views
Proving $\sum_{k=1}^na_k\sin(kx)=\sum_{k=1}^nb_k\cos(k x)$, with real $a_k$ and $b_k$, has at least one real solution
Prove that for every positive integer $n$ and any real numbers $a_1,a_2,\cdots , a_n, b_1,\cdots, b_n $, the equation
\begin{equation} \phantom{=}a_1 \sin(x)+a_2 \sin(2x)+\cdots +a_n \sin(nx) \\ = b_1 ...
- 155
0
votes
3
answers
80
views
What is the value of $[(1^ {-2/3}) + (2^ {-2/3}) + (3^ {-2/3}) + … + (1000^ {-2/3})] $? Where $[x]$ stands for the greatest integer function.
What is the value of
$$[(1^ {-2/3}) + (2^ {-2/3}) + (3^ {-2/3}) + … + (1000^ {-2/3})]?$$
where $[x]$ stands for the greatest integer function.
P. S : I ran a code on my PC to find that the ...
- 183
1
vote
0
answers
75
views
Taylor Series inverse function
Find a function represented by the Taylor series $\displaystyle\sum_{k=0}^{\infty}(-1)^k\cdot\dfrac{3^{2k+1}}{(2k+1)!}\cdot x^{2k}$.
This series looks very similar to the result for sin 3x. However, ...
- 11
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,
$$\sum_{i=1}^{n} \frac {f(x)}{(...
- 363
1
vote
1
answer
77
views
If $|f(p+q)-f(q)|\leq p/q$ for rational $p$ and $q$ (with $q\neq 0$), then $\sum_{i=0}^k |f(2^k) -f(2^i)| \leq k(k-1)/2$
If $\left|f(p+q) - f(q)\right|\leq p/q$ for all rational $p$ and $q$, with $q \neq 0$, then prove that
$$\sum_{i=0}^k \left|f\left(2^k\right) -f\left(2^i\right)\right| \leq \frac12 k(k-1)$$
My try:...
- 800
3
votes
1
answer
88
views
If $\sum_{n=1}^{\infty} \sum_{k=1}^{\infty} f_{n}(k) = \infty$ then $\sum_{k=1}^{\infty} \sum_{n=1}^{\infty} f_{n}(k) = \infty$.
Suppose $\{f_{n}\}_{n=1}^{\infty}$ be functions such that $f_{n} : \Bbb{N} \rightarrow \Bbb{R}^{+}$ for each $n$.
I was trying to prove -
If $\sum_{n=1}^{\infty} \sum_{k=1}^{\infty} f_{n}(k) = \...
- 5,042
0
votes
1
answer
77
views
Why is there a $m-1$ when approximating a lower bound of a function through summation or integral?
For a monotonically increasing f(x), why does the summation below, on the left hand side, always approximate the lower bound of the summation on the right hand side?
$\int_{k=m-1}^n f(k) \leq \sum_{...
16
votes
2
answers
1k
views
Is $\lim\limits_{k \to \infty}\left[ \lim\limits_{p \to \infty} \frac{M}{1+3+5+\cdots+ [2^{p(k-1)}-2^{p(k-2)}-2^{p(k-3)}-\cdots-1]}\right]=1$?
Firstly, my $\LaTeX$, Mathematics and English knowledge is very limited. It is extremely difficult for me to ask this question. Now I am improving myself.I hope you understand me...
Look at this ...
- 375
0
votes
0
answers
100
views
Find the function $f(x)$ under the given conditions
I need to find $f(x)$,
If $$ \frac{1/x_m}{\sum_u 1/x_u} = \frac{f(x_m)}{\sum_u f(x_u)} $$
$x_u > 0, \forall u$, $u = 0,1,2,3,...,U$, and $x_m = x_u$ if $m = u$
Could $f(x)$ be any thing other ...
- 13
2
votes
1
answer
136
views
Sum of the inverse of all positive integers which do not contain the digit 8 [duplicate]
Let $No8$ be the set of positive integers that do not contain the digit $8$. For example, $123456790 ∈ No8$ but $1234567890 \notin No8$. Show that $$\sum_{n\in No8} \frac 1n<80$$
The bound in the ...
user388085
1
vote
3
answers
548
views
Find the area of the region $y=2x$, the $x$-axis, lines $x=1$ and $x=4$
Find the area of the region $y=2x$, the $x$-axis, lines $x=1$ and $x=4$
Here's what I did:
$$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
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
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
1
vote
1
answer
42
views
What is the series of the function 3 / ( 1- x^4)
I know that $f(x) = \frac{1}{ 1-x } = \sum_{n=1}^\infty x^n$. We can find that $g(x) = \frac{1}{ 1-x^4 } = \sum_{n=1}^\infty (x^4)^n = \sum_{n=1}^\infty x^{4n}$. Does the sum converge? what is the ...
- 11
0
votes
1
answer
41
views
Directional Derivate for sums
I know the directional derivative as $D_uf(x)=\nabla f(x) . u$
But I do not know how this applies here?
- 585
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