All Questions
12
questions
2
votes
2
answers
141
views
How to choose which stronger claim to prove when proving $\sum_{i=1}^n \frac{1}{i^2} \le 2$?
I am studying an inductive proof of the inequality $\sum_{i=1}^n \frac{1}{i^2} \le 2$. In the proof, it was decided to prove the stronger claim $\sum_{i=1}^n \frac{1}{i^2} \le 2-\frac{1}{n}$, as this ...
- 3,019
1
vote
1
answer
73
views
Prove or disprove that $\sum_{k=1}^p G(\lambda^k) = ps(p)$
Prove or disprove the following: if $\lambda$ is a pth root of unity not equal to one, $G(x) = (1+x)(1+x^2)\cdots (1+x^p),$ and $s(p)$ is the sum of the coefficients of $x^n$ for $n$ divisible by $p$ ...
- 1,225
2
votes
1
answer
89
views
find a closed form expression for $\sum_{k=0}^n \left \lceil\sqrt{2k} \right\rceil, \quad n \ge 0$
The following is an exercise from my textbook. I can't really seem to find similar examples in the book and I find it a bit confusing.
I need to find a closed form expression for
$$\sum_{k=0}^n \...
user488081
1
vote
0
answers
80
views
Summing discrete coordinate lengths, a generalized n-dimensional case
A previous question pertains to a formula for the total number of points in a 3d discrete coordinate system that each total number of coordinate digit lengths $l$ can describe.
Is it possible to ...
- 33
4
votes
1
answer
70
views
A curious identity from repeated differences of integer powers
It is a well known fact that the repeated differences between $n$-powers of consecutive integers produce eventually $n!$. For example, for $n=3$ we have
\begin{eqnarray}
1, 8 , 27, 64\\
7, 19, 37\\
...
- 1,589
2
votes
1
answer
80
views
Algebra (sequence & series): $\sum_{r=1}^{n}[rx]$
How to find summation of $\sum\limits_{r=1}^{n}[rx]$ and $\sum\limits_{r=1}^{n}\left\{rx\right\}$? (where [ ] is greatest integer function & { } is fractional part)
- 41
1
vote
2
answers
93
views
Prove that $\sum_{i=0}^{m}(n+i)$ is divisible by $(m+1)$
Given a sum
$$s = \sum_{i=0}^{m}(n+i)=(n+0)+(n+1)+...+(n+m)$$
with $n, m \in \mathbb{N}$, how could I prove that $s$ is divisible by $m+1$ when $m$ is even?
I do know that
$$s = (m+1)n + \sum_{i=0}^{...
- 195
-4
votes
1
answer
102
views
Is it correct to state that a sum series converges as long as the partial sums do as well?
I can't understand that statement i heard during a math course with the fact that the partial sum's series of the natural numbers don't converge and the fact that according to Euler and Riemann the ...
1
vote
1
answer
80
views
Is this a valid mathematical induction proof?
The required is to prove that $\sum_{i=1}^n 1 = n$
So here's my attempt
Prove that it works for n = 1
$$\sum_{i=1}^1 1 = 1$$ $$ 1 = 1$$
Assume it works for k
$$\sum_{i=1}^k 1 = k$$
Show that it ...
3
votes
4
answers
442
views
Proving for all integer $n \ge 2$, $\sqrt n < \frac{1}{\sqrt 1} + \frac{1}{\sqrt 2}+\frac{1}{\sqrt 3}+\cdots+\frac{1}{\sqrt n}$ [duplicate]
Prove the following statement by mathematical induction:
For all integer $n \ge 2$, $$\sqrt n < \frac{1}{\sqrt 1} + \frac{1}{\sqrt 2}+\frac{1}{\sqrt 3}+\cdots+\frac{1}{\sqrt n}$$
My attempt: Let ...
- 1,530
2
votes
1
answer
904
views
Sums of consecutive odd integers, positive or negative
While supervising a student competition, my colleague and I ran across an interesting problem. Deobfuscated, it boils down to this
Given a limit value $M$, which integers in the range $1,\dotsc,M-1$...
- 8,738
3
votes
2
answers
205
views
A sum of difference of floors
I have the sum ( $M$ is any integer $> 1$ ):
$$
\sum_{h = 1}^{M}\left(\,\left\lfloor\, 2M + 1 \over h\,\right\rfloor
-\left\lfloor\, 2M \over h\,\right\rfloor\,\right)
$$
and looking for a way to ...
- 464