Questions tagged [summation]
Questions about evaluating summations, especially finite summations. For infinite series, please consider the (sequences-and-series) tag instead.
18,447 questions
2 votes
1 answer
123 views
How to prove this inequality $2^n\le 2t+v_2\left( \sum_{i=t}^{2^{n-1}} \binom{2^n}{2i}\binom{i}{t}\right)$ $(1\le t\le 2^{n-1}$)?
Well, when I do the polynomial problem that my teacher gave me. I've tried new way to solve the problem by using p-adic $v_2$ but cannot solve it, here is the question: For two polynomials with ...
2 votes
1 answer
171 views
Can we simplify the alternating sum $\sum_{i=0}^{n}(-1)^{i}{{n+i}\choose{i}}$?
I am doing a project and somehow the alternating sum $\sum_{i=0}^{n}(-1)^{i}{{n+i}\choose{i}}$ comes up. I am not not sure if there is any use of this sum, but just think it is an interesting sum ...
- 171
2 votes
1 answer
150 views
Limit with floor sums reminiscent of the exponent of the central binomial coefficient
This problem comes from the 1976 Putnam exam. Evaluate $$ L=\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^n \left( \left\lfloor\frac{2n}{k}\right\rfloor -2\left\lfloor\frac{n}{k}\right\rfloor \right), $$ ...
- 23
3 votes
2 answers
130 views
Prove that $\sum_{k=1}^{n} \left\lfloor \log_{2}\!\left(\frac{2n}{2k-1}\right) \right\rfloor = n$ where $n$ is a natural number.
How can I prove that $\sum_{k=1}^{n} \left\lfloor \log_{2}\!\left(\frac{2n}{2k-1}\right) \right\rfloor = n$, where $n$ is a natural number? I discovered this identity while trying to prove Prove using ...
5 votes
2 answers
197 views
Closed form for a symmetric sum of squared binomials
I am trying to find a closed form for the following sum: $$ \sum_{k=0}^{n-1} \left( \frac{1}{(k+1)(n-k)} \cdot \binom{n+1}{k+1}^2 \right) $$ What I have tried so far I tried to simplify the expression ...
- 51
2 votes
1 answer
111 views
Proof of a formula involving central binomial coefficients.
Give a combinatorial proof that \begin{eqnarray*} \sum_{i+j+k=n} \binom{2i}{i} \binom{2j}{j} \binom{2k}{k} = (2n+1) \binom{2n}{n}. \end{eqnarray*} Where did this come from ? ... In this question (...
- 37.5k
7 votes
3 answers
315 views
Telescopic series: how to identify it after breaking it down into partial fractions?
I’m trying to understand how to recognize when a series is telescoping. Consider the series $$ \sum_{n=3}^{\infty} \frac{1}{n(n-1)(n-2)}. $$ Using partial fraction decomposition, we get $$ \frac{1}{n(...
- 8,896
1 vote
1 answer
42 views
$a_1,\dots,a_n$ periodic sequences summing to $p_1,\dots,p_n$ over each of their resp. periods, then their sums synch. to some value $\leq\sum_i p_i$.
Conjecture. Let $(a_i(j))_{j \geq 0}$ be sequences of natural numbers $\geq 1$. For example $a_1 = \overline{2} = 2,2,2,2, \dots$, is the constant $2$, but $a_3 = \overline{2,1,2}$ is not. Define $B =...
- 22.7k
3 votes
0 answers
140 views
Evaluate sum $\sum_{m=0}^{n-1}\frac{\binom{2m}{m}}{2^m}$
$\newcommand{\poch}[2]{{\left(#1\right)}_{#2}}$ Consider the sum $$ S_n = \sum_{m=0}^{n-1} \frac{\binom{2m}{m}}{2^m}. $$ Using identity $$ \binom{2m}{m} = 4^m \frac{(1/2)^{\overline{m}}}{m!}, \quad \...
- 31
0 votes
3 answers
111 views
Is this a correct closed form of $\sum_{n=0}^k(-1)^nn\binom kn$?
I have the sum $$\sum_{n=0}^k(-1)^nn\binom kn$$ where I expect $k\ge 2.$ My analysis is that $$(n+1)\binom k{n+1}-n\binom kn=k\binom{k-1}n$$ and therefore the original sum evaluates as $$\sum_{n=0}^k(...
- 8,343
11 votes
2 answers
419 views
Evaluating $\sum_{m=0}^\infty \sum_{n=0}^\infty \frac{1}{(4m + 1)(4n + 1)(4m + 4n + 3)}$
I am evaluating the following integral: $$\int_0^{1} \left(\tanh^{-1}(x) + \tan^{-1}(x)\right)^2 \; dx$$ After using the Taylor series of the two functions, we get the sum: $$\sum_{m=0}^\infty \sum_{n=...
- 436
6 votes
2 answers
240 views
proving $\sum_{n=1}^{100} \frac{\sin n}{n^2} > \frac{100}{99}$
how to rigorously prove $$\sum_{n=1}^{100} \frac{\sin n}{n^2} > \frac{100}{99}? $$ Since $|\sin n| \leq 1$, $$\left| \sum_{n=1}^{100} \frac{\sin n}{n^2} \right| \leq \sum_{n=1}^{100} \frac{1}{n^2} &...
- 87
3 votes
1 answer
101 views
How to show $4^{-n} ( {2n\choose n} x + \sqrt{n} \sum_1^{n} {2n\choose n-k} \frac{\sin(2kx/\sqrt{n})}{k})$ approaches the error function for large n? [closed]
I've been asked to show $$\lim_{n \to \infty} 4^{-n} \left( {2n\choose n} x + \sqrt{n} \sum_{k=1}^{n} {2n\choose n-k} \frac{\sin\left(\frac{2kx}{\sqrt{n}}\right)}{k}\right) = \int_0^x e^{-t^2}dt$$ ...
- 369
1 vote
1 answer
119 views
Recommendation on books/notes that treat Summations rigorously [closed]
I am an undergraduate in Mathematics, almost finishing the degree. Treatment of summations (sigma notation) has always bothered me, since in most cases we can convince ourselves that their ...
- 437
1 vote
0 answers
62 views
Prove that $ \sum_{i+j=k} a_ib_j = \sum_{t=0}^k a_tb_{k-t}$
How would one prove the following result? For a given k show that: $$ \sum_{i+j=k} a_ib_j = \sum_{t=0}^k a_tb_{k-t}$$ I am not asking why this is true, this is clear. What I am asking is how would one ...
- 437