All Questions
13
questions
3
votes
1
answer
84
views
Dirichlet series of $\ln(n) \tau(n)$
I was experimenting with a technique I developed for double/multiple summation problems, and thought of this problem:
Find
$$S(p)=\sum_{n=1}^{\infty} \frac{\ln(n) \tau(n)}{n^p}$$
where $\tau(n)=\sum_{...
user1214193
1
vote
2
answers
83
views
Find all the integers which are of form $\dfrac{b+c}{a}+\dfrac{c+a}{b}+\dfrac{a+b}{c}, a,b,c\in \mathbb{N}$, any two of $a,b,c$ are relatively prime.
I have a question which askes to find all the integers which can be
expressed as
$\displaystyle \tag*{} \dfrac{b+c}{a}+\dfrac{c+a}{b}+\dfrac{a+b}{c}$
where $a,b,c\in \mathbb{N} $ and any two of $a,b,...
- 921
4
votes
1
answer
226
views
A proof of the theorem $1+\frac{1}{2}+\dots+\frac{1}{n} = \frac{k}{m}$ with $k$ odd and $m$ even
I tried to prove the following theorem :
If $n>1$ then $\displaystyle1+\frac{1}{2}+\dots+\frac{1}{n} = \frac{k}{m}$ where $k$ is odd and $m$ is even.
and I'd like to know if there is any flaw in ...
- 3,990
0
votes
2
answers
68
views
The sum of digits of a multiple of 10 converted to base 11 is still a multiple of 10?
Is it true that the sum of digits of a multiple of 10 if converted to it's base 11 form is also a multiple of 10. Is there a formal proof on this?
- 225
1
vote
1
answer
481
views
Verify my proof for "For every $n, q$, where $n,q \in Z^+, n\geq 2, \sum_{i=1}^n\frac1n\neq q$". [$n^{th}$ harmonic number is never an integer]
Background : I have little experience writing proofs. I found this problem in the book Solving Mathematical Problems by Terrence Tao. I wish to prove that $n^{th}$ harmonic number is never an integer. ...
user614767
0
votes
1
answer
62
views
Sums of positive integers $2$ apart can be every composite number and never a prime.
So what I mean is $a=n+(n+2)+(n+4)+\cdots+(n+2m)$, like 27=7+9+11, $5\ne1+3,2+4,1+3+5$ so it can't, since those are all the possibilities.
To simplify, $a=(m+1)n+2\sum_{i=1}^mi=(m+1)(m+n)$, and since ...
- 880
1
vote
2
answers
145
views
Validating a proof that powers of 2 cannot be a sum of consecutive positive integers
I purchased a book for math puzzles. It contained a puzzle summed up as some person is reading a book. They are asked to sum the pages they just read. It's either $412$ or $512$. Which is it?
I brute ...
- 79
2
votes
2
answers
332
views
Sum first $n$ squares
I am trying to find the formula for the sum of square number but I am struggle
with it. I now that it isn't complex but I am young in maths.
What I've done:
Using odd numbers to find the pattern ...
- 319
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 ...
8
votes
3
answers
1k
views
My formula for sum of consecutive squares series?
I stumbled upon a specific series, who's Sum of squares of consecutive integers equals the sum of squares of the continuation of that consecutive integers.
For exmaple, this first number in the ...
1
vote
3
answers
257
views
Mistake in proof of sum of divisors function $\sigma(n)$
The proof derives the correct result, but I cannot see how the first equality is correct.
To begin we use the formula $\sigma(n)=\sum_{d\mid n}d$
This is the first step in the proof:
$$\sum_{1\leq ...
- 2,297
7
votes
1
answer
336
views
Prove that $ \sum_{1 \le t \le n, \ (t, n) = 1} t = \dfrac {n\phi(n)}{2} $
Problem: Prove that the sum of all integers $ t \in \{ 1, 2, \cdots, n \} $ and $ (t, n) = 1 $ is $ \dfrac {1}{2} n \phi (n) $, where $ \phi $ is the Euler Totient Function.
My proof:
Define the ...
- 7,716
2
votes
1
answer
177
views
Critique on a proof by induction that $\sum\limits_{i=1}^n i^2= n(n+1)(2n+1)/6$?
I need to make the proof for this
1:$$1^2 + 2^2 + 3^2 + ... + n^2=\frac{(n(n+1)(2n+1))}{6}$$
By mathematical induction I know that,
If P(n) is true for $n>3^2$ then P(k) is also true for k=N and ...
- 653