Questions about evaluating summations, especially finite summations. For infinite series, please consider the (sequences-and-series) tag instead.

learn more… | top users | synonyms

1
vote
1answer
28 views

Find the indefinite integral $\int \left(k+2\right)^{-1}\sum_{k=0}^{\infty}\left(\frac{t^2}{1+t^2}\right)^{k+2}dt$

I need to find the indefinite integral \begin{align}I&=\int \sum_{k=0}^{\infty}\left(k+2\right)^{-1}\left(\frac{t^2}{1+t^2}\right)^{k+2}dt.\end{align} It is the result of removing the first two ...
0
votes
1answer
48 views

Equating summations of an integer partition and its conjugate

I've been given an integer partition A = (A1, A2, ... ,An) and its conjugate B = (B1, B2, ... ,Bm). Using that information, I'm tasked with proving that $$\sum_{i=0}^n (i-1)A_i = \sum_{j=1}^m B_k(B_k ...
2
votes
1answer
38 views

Characteristic polynomial forms a basis for a polynomial

This comes out of the book called Numerical analysis by Quarteroni, et al. Prove that the characteristic polynomial $l_i\in\mathbb{P}_n$ defined in (8.3) form a basis for $\mathbb{P}_n$ Background ...
0
votes
3answers
35 views

Is the following expression true or false? $\sum_{i,j=1}^n ij=2\sum_{1\le i< j\le n} ij+\sum_{i=1}^n i^2$

$\sum_{i,j=1}^n ij=2\sum_{1\le i< j\le n} ij+\sum_{i=1}^n i^2$ where n is a positive integer. I have just today learned the summation today and I still can't calculate this expression. I would be ...
1
vote
0answers
46 views

Proving a Challenging Triple Summation Identity involving Bell Polynomials

I need to prove that the following expression is true: \begin{align*} \frac{d}{dx}[fg]\sum_{c=1}^{n-1} \sum_{k=c+1}^n \sum_{j=0}^{min\{c,k-c\}} \Phi(k,c,j) f^{k-c-j} g^{c-j} \lambda(k,c,x) = \\ ...
0
votes
0answers
41 views

Physics related integrals and sums. Black Body Radiation.

How were these calculated? $$\langle E\rangle = \frac{\displaystyle\int_0^\infty Ee^{-E/kT}{\rm d}E}{\displaystyle\int_0^{\infty}e^{-E/kT}{\rm d}E}=kT$$ $$ \langle E\rangle = ...
0
votes
0answers
40 views

Convergence of sum of random variables to the normal distribution

The classical CLT speaks about the convergence of centered and normed sum of random variables to the standard normal variable. But what about the convergence of noncentered and nonnormed sum of ...
3
votes
2answers
80 views

Prove $\sum_{i=1}^n i! \cdot i = (n+1)! - 1$?

Prove the summation: $$\sum_{i=1}^n i! \cdot i = (n+1)! - 1$$ using induction. base case: $n=1$: \begin{align*} \sum_{i=1}^1 i! \cdot i &= (1+1)! - 1 \\ 1 &= 2 - 1 \\ 1 &= 1 ...
-5
votes
1answer
63 views

Prove type questions [closed]

Prove summation formula by counting a set in two ways $$ (a)\sum_{k=1}^n 2^{k-1}=2^{n}-1\,. $$ $$ (b)\sum_{k=0}^n k{n \choose x} =n2^{n-1} $$
5
votes
2answers
100 views

How do we prove the following binomial identity?

I tried to prove it by expanding the left hand side, but to no avail. Can you please explain me how to prove this statement? I'm thinking calculus(differentiation) can be used to prove this, as ...
1
vote
2answers
26 views

How telescoping sum is applied in this case?

I am reading the book Foundation of Machine Learning, and the author has many proofs for different theorems. Here is one part of a proof about Perceptron algorithm which I don't quite understand ...
0
votes
2answers
53 views

Infinite sum, sequence and series, limit

I am unable to crack the following question on sum of sequence and series your help is very much appreciated..thanks $\sum_\limits{i\geq 1} i^2x^i$
2
votes
1answer
82 views

Help Show Binomial Identity: $\sum_{j=0}^{n} {n \choose j}{m+j \choose n} = \sum_{j=0}^{n} {n \choose j}{m \choose j}2^j$ [duplicate]

I have been trying to solve this problem that I found in my old course notes for some time, but I have not been successful. Can anyone suggest a strategy or provide a hint? ...
1
vote
0answers
15 views

How to express this sum of certain non-divisors in a different form?

I need help finding a way to express this sum. $$\sum_{\substack{b\space\nmid\space x\\{\left\lfloor{\frac{x}{b}}\right\rfloor}\space\text{is even}}}^xb$$ I need to express this in a different form ...
-2
votes
3answers
72 views

Trying to prove $\sum\limits_{k=0}^{n}\binom{n}{k}=(1+1)^n$ [duplicate]

I am trying to show in the following equality that the left hand side equals the right hand side. I tried expanding out the summation but that didn't get me anywhere. Could somebody provide a hint? ...
-2
votes
1answer
36 views

If $\sum_{x \in X} f(x) < {\infty}$ then the set $\{x | f(x) \neq 0 \}$ is countable [duplicate]

Let $f:X \to R$ with $f(x) \ge 0$ for $x \in X$. Show that if $\sum_{x \in X} f(x) < {\infty}$ then the set $\{x / f(x) \neq 0 \}$ is countable Could anyone help me to show this?
0
votes
4answers
61 views

Finding the sum of products.

Consider the following two sets of consecutive integers $\{10,11,...19,20\}$ and $\{21,22,...29,30\}$. Each element of the first set is multiplied,in turn, by each element in the second set. Find the ...
0
votes
1answer
42 views

I see that this is true through computation but not proof

I am looking at the following problem to see if it is true. I have made up an example and computed it and I do see that is true. However, I would like to see how this is proved, as the proof is ...
0
votes
1answer
21 views

Problem regarding changing bounds of summation

We have the following expression: $$ A(x) = f'\sum_{c=1}^{n-1} \sum_{k=c+1}^n \sum_{j_1} \Phi(k,c,j) f^{k-c-j}g^{c-j} \sum_{l=1}^{n-1} {n \choose l}\rho(f,n-l,c-1)\rho(g,l,k-c) + g'\sum_{c=1}^{n-1} ...
0
votes
2answers
45 views

How to prove that $\sum_{k=1}^{n-1}\frac{\binom{n-1}{k-1}}{k}=\frac{2^n-1}{n}$?

My problem is to prove the identity: $$\sum_{k=1}^{n-1}\frac{\binom{n-1}{k-1}}{k}=\frac{2^n-1}{n}$$ Basically I am trying to show that the LHS is equal to the RHS. I have put this into wolfram and ...
2
votes
2answers
97 views

Solve that $\displaystyle\lim_{n\to\infty}\frac{\sum_{k=1}^n\sin\sqrt k}{n}=?$

I guess$\displaystyle\lim_{n\to\infty}\frac{\sum_{k=1}^n\sin\sqrt k}{n}=0$. If it converges to zero, it is very slow. Even I make $n=1,000,000$, the result is $-0.00112289$. Will anyone help me ...
0
votes
1answer
15 views

Proving that the point spectrum of the right shift operator on $\mathscr{l}^2(\mathbb{Z})$ is empty.

How can I prove that no series $x\in\mathscr{l}^2(\mathbb{Z})$ of the form $$\forall i\in\mathbb{Z}: x_{k-1}=\lambda x_k$$ exists other than the zero sequence? In particular I want to prove that the ...
9
votes
3answers
417 views

Infinite sums: adding terms

I would like to know where I can find a formal treatment of an idea I had, assuming it makes sense. Consider the infinite sum $$\sum\limits_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$$ and define ...
2
votes
2answers
51 views

Prove that using induction that $\binom22+\dots+\binom n2 = \binom{n+1}2$ [duplicate]

so I have this math problem where I have to prove this using induction. ...
2
votes
1answer
91 views

Proof using binomial Theorem

so I have to prove this using the binomial theorem: $$\sum_{k=0}^n{(-1)^k\begin{pmatrix}n\\k\end{pmatrix}}=0$$ I know the binomial theorem states: ...
0
votes
2answers
50 views

Limit of $\sum\limits_{k=1}^n \frac{1}{2k-1}-\frac12\ln n $

It is very well known that: $$\lim_{n\to \infty} \sum_{k=1}^n \frac{1}{k}-\ln n = \gamma$$ Is there anything known about $$\lim_{n\to \infty} \sum_{k=1}^n \frac{1}{2k}-\frac{\ln n}2 $$ or ...
0
votes
1answer
29 views

How to prove convex linear combination rule.

Let $x_i, i=1\dots n$ be elements of a convex subset $K$ of a linear space $X$ over the reals. Then any linear combination $\sum\limits_{i=1}^n a_i x_i$ such that $a_i \geq 0$ and $\sum a_i = 1$ is ...
0
votes
0answers
16 views

what is such a series called and is there a formula for its sum

I have a series of numbers whose $t^{th}$ term is defined as $S_t = (a/t)^t$ I want to know if there has been a study on sequences such as these, and if so, is there a formula for the sum of the ...
0
votes
1answer
54 views

Prove $\sum_{k=m}^n {k \choose m} = {n+1 \choose m+1}$ [duplicate]

I am trying to determine whether $\sum_{k=m}^n {k \choose m} = {n+1 \choose m+1}$, so far I am assuming that this is a false statement, but was wondering if there was a proof indicating this is a true ...
1
vote
1answer
69 views

Prove that if $n\geq1$ then $\binom{2n}{2}=$ [duplicate]

Prove that if $n\geq1$ then $$\binom{2n}{n}=\sum_{k=0}^{n}(\binom{n}{k})^2$$ This is what I have so far: By the Binomial Theorem: ...
3
votes
2answers
78 views

Prove that $\sum_{k=1}^{n} \frac{1}{k}>\ln(n+1)$ for all $n\geq1$

Prove that $$\sum_{k=1}^{n} \frac{1}{k}>\ln(n+1)$$ for all $n\geq1$ I am looking for a clear solution to this problem. I've considering trying to prove it by contradiction by starting off ...
1
vote
2answers
74 views

Show that $|x|=\frac{\pi}{2} - \frac{4}{\pi}\sum\limits_{k=0}^\infty\frac{\cos\left((2k+1)x\right)}{(2k+1)^2}$

The objective is to find the Fourier series for $|x|$ in the range $-\pi \le x \lt \pi$ so I started by finding the Fourier coefficients: ...
1
vote
0answers
58 views

Solving system of equations with summation

Is there a way to express $y_t$ in terms of $\delta$ and $p_t$ in the following system? $$ \begin{cases} & \sum\limits_{t=0}^{\infty}\delta^t\cfrac{p_t}{p_t+(1-p_t)y_t}=K \\ & ...
0
votes
0answers
41 views

Expression for moments in terms of lower moments

$$S_{n}\equiv\sum_{j=1}^{N}x_{j}^{n}$$ If the values of $\{S_{1},S_{2},...,S_{N}\}$ are known, that places N constraints on the N unknowns $\{x_{1},x_{2},...,x_{N}\}$, so we could in theory calculate ...
1
vote
2answers
32 views

Can somone help me do this double sum problem. I know how to do it manually, but I would like to know how to do it using summation formulas.

Calculating the double sum: $$\sum\limits_{i=1}^{10}\sum\limits_{j=0}^{15}(3i+2j)$$ I know how to do this manually, but I would like to know how to do it using a summation formula. Could somone ...
0
votes
2answers
34 views

Simple measurable functions properties

A simple function can be defined as $s=\sum_i^na_iA_i$, $x\in X$, where $n$ is a positive integer, $a_1,a_2,...,a_n$ are extended real numbers and $A_i$ $\subseteq X$ for every $i$. We have the ...
0
votes
1answer
43 views

Upper bound for $\sum_{i=2}^{n} 1/(i\log(i))$

I tried to find the asymtotic upper bounds for these summations $$\sum_{i=2}^{n} 1/(i\log(i)) \text{ and } \sum_{i=2}^{n} 1/(i\log(i)\log\log(i)) .$$ My guess is that they might be bounded by ...
3
votes
5answers
73 views

What is the sum of the series with binomial sequences: $\sum_{k=0}^{n} k \binom{n}{k}$? [duplicate]

compute this sum: $\sum_{k=0}^{n} k \binom{n}{k}$ I tried but I got stuck
0
votes
2answers
44 views

Complex number - sum of imaginary part

Question: Let $z = \cos \theta + i \sin \theta$. Then what is the value of $$\sum^{15}_{m=1}\text{Im}(z^{2m-1})$$ at $\theta = 2^o$ Not sure how I should attempt this question. Obviously, as ...
0
votes
0answers
43 views

Inquiry on prime counting function

One of my close friends and I have been working towards an exact prime counting function. The approach we have came up accurately produces the number of composite numbers that occur before a given ...
1
vote
1answer
26 views

Having trouble find sum of this equation which has exponents. Can someone please help me solve this

My question is: Find the value of $$\sum\limits_{j=1}^{100}(3^j+3\cdot 2^j)$$ Leave answer as powers of $2$ and $3$. I've really tried to think of a way to solve this but cant seem to find ...
0
votes
0answers
10 views

Finding integral upper bound for a sum with uni-modal function

How can I find a tight lower bound (maybe integral) for the following sum: $ F= \sum_{r=k+1}^{\infty} \frac{m}{r 2^r} {r \choose \frac{m+r}{2}}$ where $k \geq m$ . Ignore the terms under the sum ...
-1
votes
1answer
45 views

Prove $\sinh1\Bigl(1+2{\sum}_{n=1}^{\infty}\frac{(-1)^{n}}{(n\pi)^2+1}\Bigr)=1$ for $-1\lt x \lt 1$ [duplicate]

This is a follow up topic from this previous post. The Fourier series for $f(x)=e^x$ is $$f(x)=e^x=\sinh1\left(1+2\sum_{n=1}^{\infty}\frac{(-1)^n}{(n\pi)^2+1}\left(\cos(n\pi x)-n\pi\sin(n\pi ...
1
vote
2answers
57 views

Exact value or lower bound for $\sum\limits_{k=m}^{\infty} \frac{\lambda^{k}}{k!}\sum\limits_{r=m}^{k} \frac1{r 2^r} {r \choose (m+r)/2}$

How can I find the exact value or a tight upper bound when $m \to \infty$ for the sum $$ F= \sum_{k=m}^{\infty} \frac{e^{-\lambda}\lambda^{k}}{k!}\sum_{r=m}^{k} \frac{m}{r 2^r} {r \choose ...
0
votes
1answer
17 views

How to label table cells and create the formula for a summation?

I want to label each cell in a table and then create the formula of a summation over each column. I'm new to math and not really sure if my syntax is correct. Espacially the labeling over 3 layers ...
-1
votes
2answers
58 views

Proving convergence of infinite sum $\sum_{i=1}^{\infty} \frac{A}{(1+r)^i} = \frac{A}{r}$ [duplicate]

I have to show that $\sum_{i=1}^{\infty} \frac{A}{(1+r)^i} = \frac{A}{r}$. Where $ -1 < r < 1$. I tried putting it like: $$\sum_{i=1}^{\infty} \frac{A}{(1+r)^i} = \frac{A}{r+1} + ...
0
votes
1answer
38 views

For $n \in \mathbb{N}$, find and prove a formula for $\sum_{i=1}^n \frac{1}{i(i+1)}$, plus related question.

I was fairly easily able to obtain and prove this formula for the sum: $$S(n)=\frac{n}{n+1}$$ by typical means of computing the partial sums, observing the pattern, and proving by induction. My ...
1
vote
2answers
40 views

Find sum of $\sum_{i=1}^{8}$ $\sum_{j=1}^{10}$ $(2i+2j)$

The question is how to find $\sum_{i=1}^{8}$ $\sum_{j=1}^{10}$ $(2i+2j)$. So I worked inside out, and split the inside sum into two as such: $2\sum_{j=1}^{10} i + 2\sum_{j=1}^{10} j$ Second one ...
7
votes
3answers
79 views

Prove $2^{2n} = \sum_{k=0}^{n}\binom{2n+1}{k}$

I'm trying to prove the following equation above. So far I have: \begin{align} 2^{2n} &= (1+1)^{2n}\\ &= \sum_{k=0}^{2n}\binom{2n}{k}1^k1^{n-k} = \sum_{k=0}^{2n}\binom{2n}{k} & \text{(By ...
5
votes
1answer
94 views

How did this result come about?

I was reading Chebyshev polynomials Wiki page and I could not understand one thing $$ T_n(x) = x^n \sum_{k=0}^{\left \lfloor \frac{n}{2} \right \rfloor} \binom{n}{2k} \left (1 - x^{-2} \right )^k ...