Questions about evaluating summations, especially finite summations. For infinite series, please consider the (sequences-and-series) tag instead.
1
vote
0answers
31 views
Closed form for a summation
Is there a 'closed form' for the following summation?
$$S_k=\sum_{i=0}^{m-1}\binom{mk-ik}{k}(mk-i)$$
I evaluated that $S_1=\frac{m(m+1)(2m+1)}{6}$ and $S_2=\frac{m(m+1)(7m^2-1)}{6}$.
0
votes
0answers
21 views
Finding solutions to the equation
I want to find possible solution satisfying both the equation:
$\sum_{i=1}^{n} f_i^{2} = n$
$\sum_{i=1}^{n} f_i=0$
As the number of equations less than number of variables can we just comment on ...
2
votes
1answer
36 views
Closed Form of a Particular Sum
Does anyone have any ideas on how to find a closed form for the following expression? It comes up when trying to bound a particular integral. The sum is:
$$\sum_{n=0}^{\infty} ...
0
votes
0answers
46 views
Can anyone see why this lemma is true? Seems very confusing!
Given a function f, we write $\bar{f}(t) = sup_{u \leq t} f(u).$
Lemma: Let $s_{0},.....,s_{T}$ be real numbers and $h:\Re \longrightarrow \Re.$ Then
$$\displaystyle ...
3
votes
2answers
68 views
Prove or disprove: $\sum\limits_{i=1}^n i^2 = O(n^2) $
Prove or disprove:
$$\sum_{i=1}^n i^2 = O(n^2) $$
If we want to prove this, find some summation that we know the $ O(n)$ runtime for, and is $ O(n^2) $ or smaller.
Otherwise, we could disprove ...
2
votes
3answers
55 views
problem with induction?
I am a bit new to logical induction, so I apologize if this question is a bit basic.
I tried proving this by induction:
$$\left(\sum_{k=1}^nk\right)^2\ge\sum_{k=1}^nk^2$$
Starting with the base ...
0
votes
1answer
29 views
Binomial sum of derivatives
I would like to know the result of the following sum:
$$\sum_{p=0}^m \binom{m}{p}(-1)^{p-1}\frac{\partial^{p-1}}{\partial x^{p-1}}f(x)\cdot(-1)^{m-p-1}\frac{\partial^{m-p-1}}{\partial ...
1
vote
1answer
31 views
Summation involving Hypergeometric exponential and factorial
I am not able to solve the following sum. Can you please provide any hints ?
$$
\sum_{y=1}^\infty {}_1F_1(1-y;2;-\pi\lambda c) \frac{\lambda^y}{y!}
$$
Thank you for your time
0
votes
0answers
24 views
Summation involving Laguerre polynomial
Is there any chance of expressing the above summation with simple elementary or special functions?
$$
\sum_{N=1}^\infty \frac {\lambda^N} {N!}\frac {1}{N}L_{N-1}^{(1)}(-\pi \lambda c)
$$
Thanks in ...
0
votes
1answer
44 views
Could $\sum e^{a_i}$ be simplified? Does it have an identity?
$\sum_{i=1}^n e^{a_i}$ (where $a_i \in \mathbb R$) is expensive for large $n$ (a sum and $n$ exponential operations). I was wondering if there is any way for simplifying this?
1
vote
1answer
62 views
The sum $\sum_{k=1}^{n-1}{10k+5\over (k+1)!2^k}$
I'm solving a problem, and I'm getting the following sum as the solution. $$\displaystyle\sum_{k=1}^{n-1}{10k+5\over (k+1)!2^k}$$
Wolfram Alpha says this can be simplified. How do I arrive at the ...
1
vote
3answers
68 views
Using induction to verify a statement
I have to prove that this statement is true.
For $n = 1, 2, 3, ...,$ we have $ 1² + 2² + 3² + ... + n² = n(n + 1)(2n + 1)/6$
Basically I thought I'd use induction to prove this. Setting $n = p+1$, I ...
4
votes
1answer
105 views
Estimating the sum $\sum_{k=2}^{\infty} \frac{1}{k \ln^2(k)}$
By integral test, it is easy to see that
$$\sum_{k=2}^{\infty} \frac{1}{k \ln^2(k)}$$
converges. [Here $\ln(x)$ denotes the natural logarithm, and $\ln^2(x)$ stands for $(\ln(x))^2$]
I am ...
3
votes
3answers
45 views
What is the sum of this infinite series? Which one is it, Taylors? Binomial?
I am trying to figure which formula to use for this one.
$$\displaystyle\sum\limits_{x=-1}^{-\infty} -x(1-y)p(1-p)^{-x}+\sum\limits_{x=1}^\infty xyp(1-p)^x$$
where $0<y<1$, and $0<p<1$.
...
7
votes
2answers
111 views
Proving that $\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\dots+\frac{1}{\sqrt{100}}<20$
How do I prove that:
$$\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\dots+\frac{1}{\sqrt{100}}<20$$
Do I use induction?
1
vote
3answers
57 views
How to sum numerator and denominator of a fraction?
I want to do sum over this. Can apply the summation to top and bottom separately?
$$\sum\limits_{i=1}^{n} \frac{-a(x_i-\mu)^2}{x_i}$$
...
8
votes
3answers
95 views
Help me prove this inequality :
How would I go about proving this?
$$ \displaystyle\sum_{r=1}^{n} \left( 1 + \dfrac{1}{2r} \right)^{2r} \leq n \displaystyle\sum_{r=0}^{n+1} \displaystyle\binom{n+1}{r} \left( \dfrac{1}{n+1} ...
1
vote
2answers
66 views
question on summation?
Please, I need to know the proof that
$$\left(\sum_{k=0}^{\infty }\frac{n^{k+1}}{k+1}\frac{x^k}{k!}\right)\left(\sum_{\ell=0}^{\infty }B_\ell\frac{x^\ell}{\ell!}\right)=\sum_{k=0}^{\infty ...
10
votes
6answers
113 views
A limit on binomial coefficients
Let $$x_n=\frac{1}{n^2}\sum_{k=0}^n \ln\left(n\atop k\right).$$ Find the limit of $x_n$.
What I can do is just use Stolz formula. But I could not proceed.
0
votes
2answers
36 views
question about summation?
are thier any general rule to find
$???\leqslant \sum_{n=t}^{m}f(n)\leqslant ???$
when m and t $\in $ R
i hope to find the answer
and thanks for all
0
votes
4answers
53 views
Integration problems
Can anyone help me with these:-
(a)Prove by induction: $\displaystyle\sum_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$
(b) By explicitly calculating upper and lower Riemann sums on a uniform partition and
...
3
votes
0answers
26 views
How to prove this Stirling related equation
Here is what I need to prove, but have no idea were to start. I know there is some connection with the Stirling theorem.
$$
\sum_{i=0}^{d}\binom{m}{i} \leq \left ( \frac{em}{d} \right )^{d}
$$
I ...
12
votes
3answers
147 views
$\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$
Is it possible to calculate the following infinite sum in a closed form? If yes, please point me to the right direction.
$$\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$$
2
votes
2answers
32 views
Simplify summation with factorial and binomial coefficients
I would like to know how to simplify the following summation:
$$\sum_{p=0}^n\quad n!\frac{(2p)!}{(p!)^2}\frac{(2(n-p))!}{((n-p)!)^2}$$
Which rules should I use to simplify it?
Thanks!
2
votes
1answer
48 views
Sum of squared/cubed combinations
I was wondering if there is a closed formula for sum of cubed combinations. More precisely, I'd like to compute $$\sum_{k=1}^n \left ( \begin{array}{c}n\\k\end{array}\right )^3$$
Obviously, without ...
2
votes
0answers
38 views
Sum of squared/cube combinations [duplicate]
I was wondering if there is a closed formula for sum of cubed combinations. More precisely, I'd like to compute $$\sum_{k=1}^n \left ( \begin{array}{c}n\\k\end{array}\right )^3$$
Obviously, without ...
0
votes
1answer
32 views
How to maximize $n!\sum^n_{k = 0}\frac{a^k(1+(-1)^{n-k})}{k!(n-k)!} \pmod{a^2}$ for a given $a$?
Short Version of the Question:
How do I maximize the value of $n!\sum^n_{k = 0}\frac{a^k(1+(-1)^{n-k})}{k!(n-k)!} \pmod{a^2}$ for a given $a$?
Long Version of the Question:
I'm currently attempting ...
3
votes
1answer
60 views
Proof the following trig series
Prove that
$$\frac{ \sin x}{ \cos x}+\frac{\sin2x}{\cos^{2}x}+\frac{\sin3x}{\cos^{3}x}+\cdots+\frac{\sin nx}{\cos^{n}x}=\cot x-\frac{\cos(n+1)x}{\sin x \cos^{n}x}$$
I am not necessarily looking for a ...
4
votes
3answers
69 views
A binomial identity from Mathematical Reflections
Here is the problem:
Let $m,n$ be positive integers with $n>m$. Prove that
$\displaystyle\sum_{k=0}^{n} (-1)^{k}\binom{n}{k}\binom{m+n-2k}{n-1}=\binom{n}{m+1}$
This problem is O243 of ...
6
votes
4answers
110 views
Sum of kth roots ($\sum\sqrt[k]{m}$)
I'm trying to find an asymptotic to $$S(n) = \sum_{k=1}^n\sqrt[k]{m}$$ From computational tests, it seems to grow nearly as slowly as $n$. However even $$\sum_{k=1}^\infty\sqrt[k]{m}-1$$ diverges (for ...
4
votes
3answers
48 views
Summation using binomial coefficient
I am trying to calculate the following using binomial coefficients and summation, but my memory is standing in the way:
$$
\sum_{k=1}^n {k} * 2 ^ {k - 1}
$$
Thanks!
With great help I got to:
$$ ...
1
vote
2answers
36 views
$x$ as the shortest alternating sum of $1 \ldots n$
If I have an positive integer $x \in \mathbb{N}$ and I have $Z = \sum_{i = 0}^{n}{i}$ such that $Z \geq x$ and $Z - x \equiv 0 \bmod 2$ and $n$ is the smallest such integer it is possible to create ...
0
votes
0answers
72 views
Simplifying $\frac{1}{n}\sum_{k=1}^n f(\frac{1}{k})$
Suppose that $$\displaystyle \forall x\in \mathbb{R}_+^* \quad f(x)=\frac{x^2-1}{4}-\frac{\ln(x)}{2}.$$
How can I simplify this: $$I(n)=\frac{1}{n}\sum_{k=1}^n f\left(\frac{1}{k}\right)$$
and prove ...
2
votes
1answer
34 views
Hermite's equation of order $\alpha$
Show that the general solution of Hermite's equation of order $\alpha$:
$${y}''-2x{y}'+2\alpha y=0$$
$$is$$
$$y(x)=c_{0}y_{1}(x)+c_{1}y_{2}(x)$$
where $y_{1}(x)$ and $y_{2}(x)$ are power series ...
10
votes
3answers
133 views
Combinatorial proof of $\sum^{n}_{i=1}\binom{n}{i}i=n2^{n-1}$.
Prove
$$\sum^{n}_{i=1}\binom{n}{i}i=n2^{n-1}$$
I can't find counting interpretations for either of the sides. A hint of "if $S$ is a subset of $\{1, . . . , n\}$ and $S^\prime$ is its complement ...
0
votes
1answer
16 views
Absolute convergence.
Determine if absolutely convergent or not; Justify.
$$\sum_{n=1}^\infty (-1)^n n^2 3^{1-n} x^n \text{ s.t }|x|<3$$
if we take the abs value of $(-1)^n$ we are left with $n^{2} 3^{1-n} x^{n}$ now ...
0
votes
1answer
14 views
Clues to prove average in T is minor or equal than average in a smaller inner interval.
Suppose I want to prove (or disprove) this assertion
Let $f$ be a discrete function, $T,h,k$ are constants
So these terms are averages over $T$ and over $h$
$\sum\limits_{i=0}^{T}\frac {f(i)}{T}$ ...
2
votes
5answers
111 views
Proving $\sum_{k=1}^n{k^2}=\frac{n(n+1)(2n+1)}{6}$ without induction [duplicate]
I was looking at: $$\sum_{k=1}^n{k^2}=\frac{n(n+1)(2n+1)}{6}$$
It's pretty easy proving the above using induction, but I was wondering what is the actual way of getting this equation?
5
votes
2answers
57 views
Binomial probability with summation
Show that
$$\sum_{k=0}^{m} \frac{m!(n-k)!}{n!(m-k)!} = \frac{n+1}{n-m+1}$$
Attempt:
It becomes:
$$\sum_{k=0}^{m } \frac{\binom{m}{k}}{\binom{n}{k}}$$
Telescoping, pairing, binomial theorem don't ...
3
votes
3answers
46 views
Help finding the limit of a sum
Hi I'm trying to find the following limit:
$$\lim_{n \rightarrow \infty} \frac{1}{n} \sum_{j=1}^{ n } (1 - e^{\frac{-jt}{n}} )$$
expressed as a funciton of t. You may even be able to get it from ...
1
vote
3answers
129 views
Evaluate a sum with binomial coefficients
$$\text{Find} \ \ \sum_{k=0}^{n} (-1)^k k \binom{n}{k}^2$$
I expanded the binomial coefficients within the sum and got $$\binom{n}{0}^2 + \binom{n}{1}^2 + \binom{n}{2}^2 + \dots + \binom{n}{n}^2$$
...
8
votes
2answers
118 views
A sum with binomial coefficients
Show that $$\sum_{k=0}^{n}(-1)^k\binom{n}{k}(n-2k)^{n+2}=\frac{2^{n}n(n+2)!}{6}.$$
2
votes
1answer
40 views
Trigonometric Integration + Series
I am doing an integration question:
$$\int \frac{1-\cos^{2m}x}{1-\cos^2x}$$
So I have to show that $$\frac{1-\cos^{2m}x}{1-\cos^2x}=1+\cos^2x+\cos^4x+...+\cos^{2(m-1)}$$
How can I do that?
5
votes
4answers
114 views
Summation of logs
Are there any useful identities for quickly calculating the sum of consecutive logs? For example $\sum_{k=1}^{N} log(k)$ or something to this effect. I should add that I am writing code to do this (as ...
6
votes
1answer
66 views
Closed form for $\sum_{n=1}^\infty\frac{(-1)^n n^a H_n}{2^n}$
Is there a closed form for the sum $$\sum_{n=1}^\infty\frac{(-1)^n n^a H_n}{2^n},$$ where $H_n$ are harmonic numbers: $$H_n=\sum_{k=1}^n\frac{1}{k}=\frac{\Gamma'(n+1)}{n!}+\gamma.$$
This is a ...
9
votes
3answers
100 views
Closed form for $\sum_{n=1}^\infty\frac{(-1)^n n^4 H_n}{2^n}$
Please help me to find a closed form for the sum $$\sum_{n=1}^\infty\frac{(-1)^n n^4 H_n}{2^n},$$ where $H_n$ are harmonic numbers: $$H_n=\sum_{k=1}^n\frac{1}{k}=\frac{\Gamma'(n+1)}{n!}+\gamma.$$
7
votes
2answers
97 views
Efficient computation of $\sum_{k=1}^n \lfloor \frac{n}{k}\rfloor$
I realize there is probably not a closed form, but is there an efficient way to calculate the following expression?
$$\sum_{k=1}^n \left\lfloor \frac{n}{k}\right\rfloor$$
I've noticed $$\sum_{k=1}^n ...
1
vote
0answers
58 views
Bernoulli formula
The sum:
$$S_m(n) = 1^m + 2^m + 3^m + 4^m + 5^m...$$
Can be calculated by this formula, called the "Bernoulli formula" in wikipedia
$$S_m(n) = \frac{1}{m+1}\sum_{k=0}^m {m+1\choose k}B_k n^{m+1-k} ...
1
vote
1answer
29 views
Show the infinite sum is an integral
Here is a prelim question that I have not been able to solve:
Show that if $a>1$, then
$$
\sum_{n=1}^\infty n^{-1/2}a^{-n} = 1/\sqrt{\pi}\int_0^\infty \frac{y^{-1/2}}{ae^y-1}dy
$$
Thanks so ...
0
votes
2answers
66 views
I need help with proofs using mathematical Induction
I need help with this problem: $2+7+12+17+...+(5n-3)=(\frac{n}{2})(5n-1)$
