All Questions
9
questions
6
votes
1
answer
228
views
An alternating sum
I ran into an alternating sum in my research and would like to know if the following identity is true:
$$
\sum_{i = 0}^{\left\lfloor \left(n + 1\right)/2\right\rfloor} \frac{\left(n + 1 - 2i\right)^{n ...
- 147
2
votes
1
answer
77
views
equality involving sums
Let $n\in\mathbb{Z}^+.$ Prove that for $a_{i,j}\in\mathbb{R}$ for $i,j = 1,\cdots, n,$
$$\left(\sum_{i=1}^n \sum_{j=1}^n a_{i,j}\right)^2 + n^2\sum_{i=1}^n\sum_{j=1}^n a_{i,j}^2 - n\sum_{i=1}^n \left(\...
- 1,225
1
vote
0
answers
45
views
How to esimate $X^{c}/c - \sum_{1 \leq t \leq X} t^{c-1}$
Let $c > 0$. I am trying to obtain an upper bound for
$$
|X^{c}/c - \sum_{1 \leq t \leq X} t^{c-1}|.
$$
I am sure this is pretty small as $\int_{1}^X t^{c-1} dt = X^c/c - 1$.
What is the best ...
- 2,913
1
vote
3
answers
668
views
In showing integer sum $(1+2+3+...+n)$ by l'Hopital rule why they take lim as $r$ approaches to $1$ in one of steps? Why $1$?
So why lim as $r \to 1$ (why $1$?) Here's the method:
- 365
4
votes
3
answers
158
views
Prove that $\sum_{n = 1}^{n = s} n^2 \ne \sum_{n = t}^{n = u} n^2$
where $t > s$ and $s,t,u,n$ are positive integers. The inequality is my claim. I arrived at this as I was tinkering with stuff. I tried various things but no luck.
- 2,677
1
vote
2
answers
155
views
Can we prove summation formula for the first $n$ terms of natural numbers through calculus? [closed]
Can we prove summation formula for the first $n$ terms of natural numbers through calculus?
What about the summation of first $n$ numbers of the form $a^k$ and other summation formulas like sum of a ...
- 2,411
22
votes
5
answers
2k
views
Show that this sum is an integer.
I have to show that
$$g\left(\frac{1}{2015}\right) + g\left(\frac{2}{2015}\right) +\cdots + g\left(\frac{2014}{2015}\right) $$
is an integer. Here $g(t)=\dfrac{3^t}{3^t+3^{1/2}}$.
I tried to solve ...
- 1,156
1
vote
0
answers
88
views
Is there a formula for $1+\sqrt{2}+\sqrt{3}+\cdots+\sqrt{N}$? [duplicate]
Is there a known formula to the sum
$$1+\sqrt{2}+\sqrt{3}+\cdots+\sqrt{N}$$
where $N$ is some natural number? Thanks
- 27
5
votes
1
answer
2k
views
How many decimal representations are possible for the number 1
I know that there at least two
$0.\overline{9}$ and 1
Is there a neat and more concrete way to understand this problem.
- 97