All Questions
26
questions
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