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

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5
votes
5answers
272 views

Prove that the sequence with $T(0)=1$ and $T(n) = 1 + \sum_{j=0}^{n-1}T(j)$ is given by $T(n)=2^n$

$T(0)=1 \\ T(n) = 1 + \sum_{j=0}^{n-1}T(j) \\ $ Show that $T(n) = 2^n$. I know how to prove this by induction, but I would like to know how to show this using first principles. Edit: The way I want ...
2
votes
2answers
31 views

Help in explaining this sigma notation breakdown

I will appreciate some breakdown help which explains each step in the picture below to the last expression and the rules that applied to the changes. I am new to Sigma notations and thus confused.
2
votes
2answers
47 views

Help finding a summation using CAS

While approximating an integral by midpoint rule, I ended up with $$\iint_R f(x, y)\hspace{1mm}dA\approx \dfrac{1}{n^2}\sum_{i=0}^{n-1}\sum_{j=0}^{n-1} f\left[\dfrac{1}{2n}+\dfrac{i}{n},\hspace{3mm} ...
1
vote
1answer
28 views

Summation of series- substitution

If we have $\sum_{n=0}^{\infty}nf(n)=C, C\ne0\tag 1$, C is a constant, can we find a closed form for f(n)?. NB : Given condition is that $\sum_{n=0}^{\infty}f(n)$ converges to a constant value $K$ ...
-4
votes
2answers
111 views

What does this infinite sum converge to?: $\sum_{n=1}^\infty \dfrac{1}{n^k} = \dfrac{1}{1^k} + \dfrac{1}{2^k} + \dfrac{1}{3^k} + …$

$$\sum_{n=1}^\infty \dfrac{1}{n^k} = \dfrac{1}{1^k} + \dfrac{1}{2^k} + \dfrac{1}{3^k} + \dfrac{1}{4^k} + \dfrac{1}{5^k} + ...$$ I've found that: when $k=1$, it diverge to infinity when $k=2$, it ...
1
vote
0answers
68 views

Sum of a Sequnce

How to find this sum $$ \sum_{i+j+k=n} \ G_i \cdot G_j \cdot G_k \ for \ all \ i,j,k > 0, $$ $$ G_i = i \cdot F_i, $$ where Fi - ith number Fibonacci, F0=0, F1=1
1
vote
2answers
71 views

Summation of infinite series

If we know the series sum given below converges to a value $C$(constant) $$\sum_{n=0}^{\infty}a_n =C \tag 2$$ Can we generate following in terms of C. values of $a_n$ will tend to zero as n goes to ...
3
votes
3answers
57 views

Sum of products of binomial coefficients

In a proof I've come across the following identity: $$ \sum_{l=0}^{n-j} \binom{M-1+l}{l} \binom{n-M-l}{n-j-l} = \binom{n}{j} $$ I see that it's right, when plugging in numbers, but I don't see the ...
0
votes
1answer
56 views

Brackets with a summation

$$(\sum_{i=1}^6 x_i + 9) $$ Hey guys! For above, do you think the $9$ would be part of the summation (i.e $x_1+x_2+x_3+x_4+x_5+x_6+54$) or would it be separate (i.e. $x_1+x_2+x_3+x_4+x_5+x_6+9$) ...
0
votes
3answers
29 views

Finding the sum of a sequence of terms

$$1/1(2) - 1/3(2^3) + 1/5(2^5) - 1/7(2^7)$$ This is equal to $$\sum_{n=0}^\infty(1/2)^{2n+1}(-1)^n/(2n+1)$$ Differentiating this leads to: $$\sum_{n=0}^\infty(-1/4)^n$$ Which is equal to $4/5$ Thus, ...
0
votes
0answers
56 views

Sigma Notation Inequality

Given two sets of nonnegative real numbers: $$\{a_1, a_2, ..., a_N\}, \{b_1, b_2, ..., b_N\}$$ Are there any conditions for which the following inequality is true? $${1\over N} \sum_{i=1}^N ...
1
vote
1answer
58 views

Interesting combinatorial identities

Let $n$ be a strictly positive integer and let $j=0,\dots,n-1$. By using Mathematica I managed to guess the following identities: \begin{eqnarray} \sum\limits_{m=0}^{n-j-1} \binom{n-m-1}{j} ...
1
vote
7answers
121 views

Error in proving of the formula the sum of squares

Given formula $$ \sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6} $$ And I tried to prove it in that way: $$ \sum_{k=1}^n (k^2)'=2\sum_{k=1}^n k=2(\frac{n(n+1)}{2})=n^2+n $$ $$ \int (n^2+n)\ \text d ...
3
votes
3answers
123 views

How to count the $r$-tuples of subsets of $\{1,\dots,n\}$ that are cyclically disjoint?

I want to count the following, $$\#\{S_1,S_2,\dots, S_r\subseteq[n]\;|\; S_i\cap S_{i+1}=\emptyset \text{ for } 1\leq i\leq r-1 \mbox{ and } S_1\cap S_r=\emptyset\}=A_{n,r},$$ Then $A_{n,1}=2^n$, ...
5
votes
0answers
59 views

Subset Sum Problem Variation?

There are $100$ cards with a unique number from $1$ to $100$ written over them. How many ways can someone pick exactly $5$ cards where the numbers on them sum to $100$? I am not sure but this could ...
1
vote
1answer
56 views

The Summation of x(log(log(x))

So I would like to know if it is possible to express this summation in terms of $n$: $$\sum_{x=2}^n x\log(\log(x))$$ For example the summation below is equivalent to $\frac12 n (n+1)$ in terms of ...
10
votes
2answers
110 views

Find a closed form for this infinite sum

How to find a closed form for the expression?? $$ 1+\frac 1 2 +\frac{1 \times2}{2 \times 5}+\frac{1 \times2\times 3}{2 \times5\times 8}+\frac{1\times 2\times 3\times 4}{2 \times 5\times 8\times ...
1
vote
1answer
28 views

summation of a series in which each term is product of nth term of two sequence

Is it possible to find the sum $\Sigma_{x=1}^n ((2x)(4x+1))$? If yes then can somebody please explain for me the formula?
1
vote
3answers
487 views

Simplifying this sigma notation [closed]

Is there any way I can simplify this sigma notation? $$\begin{align*} \sum_{k=1}^m(5^k) \end{align*}$$
0
votes
1answer
39 views

Trigonometric series problem: finding a second valid solution.

Given that I can do part of this question so here goes: Substituting $\theta=\frac{1\pi}{11}$ into LHS of given expression gives $$\cos\frac{1\pi}{11} + \cos\frac{2\pi}{11} + \cos\frac{3\pi}{11} ...
3
votes
0answers
90 views

conjecture about prime numbers and distance between them

is there a name for this conjecture? Conjecture: given $p_n$ a prime number sequence where $p_1=2,p_2=3,\cdots$, then for all $n\in\mathbb{N}^*$ and $k\in\mathbb{N}$, holds that $\displaystyle ...
0
votes
2answers
34 views

Problem involving summing exponential series:

I can show the first part (i) (a), but the second part (b) i think it should be $S=\infty$ since the denominator is zero with that value of $\theta$. However, this is not the answer, any ideas? ...
0
votes
1answer
26 views

Summation of infinte series

Sir, I have three infinite summation $A =J_1 \sum_{n=2}^\infty (n-1) f(n-2) \tag 1$ , $B =\sum_{n=0}^\infty f(n) \tag 2$ and $C =J_2\sum_{n=1}^\infty f(n-1) \tag 3$, with ...
1
vote
2answers
44 views

How to show: $\sum_{k=1}^{\infty}\sum_{n=1}^{k}P(X=k)=\sum_{n=1}^{\infty}\sum_{k=n}^{\infty}P(X=k)$

Can anyone explain why this equation is true. $$\sum_{k=1}^{\infty}\sum_{n=1}^{k}P(X=k)=\sum_{n=1}^{\infty}\sum_{k=n}^{\infty}P(X=k).$$
0
votes
2answers
33 views

How do I expand this summation?

So I just started doing these today and this has me stuck (It's a beginner question and i'm upset I'm stumped). So I have $\sum_{k=1}^{4}9k\sin(\frac{k\pi}{2})$ which I turn into ...
0
votes
1answer
88 views

Evaluate $\sum_{k=0}^{511}\frac{\sin\frac\pi{2^{11}}}{\sin\frac{(4k+1)\pi}{2^{12}}\sin\frac{(4k+3)\pi}{2^{12}}}$

I need to evaluate $$\sum_{n=0}^{511}\frac{\sin\frac\pi{2^{11}}}{\sin\frac{(4n+1)\pi}{2^{12}}\sin\frac{(4n+3)\pi}{2^{12}}}$$ Please give me some hint! The final answer is $2^{10}$. By CuriousGuest's ...
0
votes
1answer
26 views

Question related to the sigma during expectation in probability [duplicate]

How is this below possible: $ \sum_{i=1}^\infty 1/i = \infty $
0
votes
2answers
67 views

Find a sum of $\sum_{n=1}^{\infty}(-1)^{n+1}\frac{ch(n)}{3^n}$

Find a sum of $$\sum_{n=1}^{\infty}(-1)^{n+1}\frac{ch(n)}{3^n}$$ Could you give some some hint or some way to start this? I have tried representing ch(n) through its definition with e, but I ...
1
vote
1answer
71 views

Multiple sum involving binomial factors

Let $n$ and $m$ be positive integers and let $0 \le j \le n-m-1$. Show that: \begin{align} \sum\limits_{l=m}^{n-j-1} \binom{n-l-1}{j} \binom{l}{m} \binom{n+l}{j} &=\sum\limits_{p=0}^j ...
1
vote
2answers
38 views

Integral of a sum of complex exponentials

Let $$\hat{\varphi_n}(t)=\frac{1}{n}\sum_{j=1}^n{exp(i{t}Y_j)}\quad(t\in\mathbb{R})$$ denote the empirical characteristic function of the residuals $Y_j\,=\,S_n^{-\frac{1}{2}}(X_j-\bar{X}_n),\quad ...
5
votes
2answers
139 views

Inequality with $\frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{2n}$

Inspired by this recent question, I suggest this. Let $n=2,3,4, \ldots .$ Then $$ \frac{7}{12} < \cfrac 1 {1 + \cfrac {1^2} {1 + \cfrac {2^2} {\ddots + \cfrac \vdots { 1 + \, {n^2} \,}}}} \leq ...
0
votes
1answer
40 views

Symmetric Sum Notation

From here is the following excerpt Suppose one is given a homogeneous symmetric polynomial $P$ and asked to prove that $P(x_1, \ldots , x_n) ≥ 0$ How should one proceed? Our first step is ...
2
votes
1answer
81 views

A double sum with combinatorial factors

Let $n$, $p$ and $j$ be integers. As a byproduct of some other calculations I have discovered the following identity: \begin{equation} \sum\limits_{p=0}^{j} \sum\limits_{p_1=0}^j \binom{p+p_1}{p_1} ...
5
votes
8answers
200 views

Show that the inequality holds $\frac{1}{n}+\frac{1}{n+1}+…+\frac{1}{2n}\ge\frac{7}{12}$

We have to show that: $\displaystyle\frac{1}{n}+\frac{1}{n+1}+...+\frac{1}{2n}\ge\frac{7}{12}$ To be honest I don't have idea how to deal with it. I only suspect there will be need to consider two ...
1
vote
1answer
34 views

How to simplify recursive eq?

I know how to programatically calculate this, but im not sure how it can be simplified for documentation. Can someone help? $R = (X\cdot 1) + (X\cdot 2) + (X\cdot 3) + (X\cdot 4) + (X\cdot 5) + ...
4
votes
4answers
764 views

Why don't we indicate the variable to summed as we do for integrals?

When integrating over a certain variable $x$, we make sure to end the integral with $dx$, like so: $$\int_{1}^{\infty}\frac{1}{x^2}dx$$ The reason for this of course becomes more clear as one goes ...
3
votes
1answer
39 views

How to simplify a sum with binomial coefficients multiplied by $k^3/2^k$?

The sum is $$\sum_{k=5}^{\infty}\binom{k-1}{k-5}\frac{k^3}{2^{k}} $$ The first thing I thought of was the binomial coefficient. So I re-indexed it ...
0
votes
1answer
19 views

A problem on orthonormality of a set of complex functions

The following is a problem of an undergraduate exam test:
1
vote
3answers
94 views

Simplifying $\displaystyle\sum_{k=0}^{20}(k+4)\binom{23-k}{3}$

In trying to simplify my answer to a problem posted recently, I am trying to show that $\displaystyle\sum_{k=0}^{20}(k+4)\binom{23-k}{3}=8\binom{24}{4}$. I know that ...
0
votes
3answers
53 views

$\sum_{(p,q) \in {\mathbb{N}^*}^2 and p \land q =1} \frac{1}{p^2 q^2} = \frac{5}{2}$ proof? [closed]

Can you give me a very precise demonstration of this result please because it's very difficult for me to understand the demonstration on the pic :( $$ \sum_{(p,q) \in {\mathbb{N}^*}^2 \text{, } p ...
0
votes
1answer
25 views

Simplify $S=\sum_{i=0}^{k}a_i (2n)^{2i+1}$

Can someone simplify this expression (or compute its supremum)? Thanks so much. $$S=\sum_{i=0}^{k}a_i (2n)^{2i+1}$$ where $a_i>0$ and $k>1$, and $\sum_{i=0}^{k}a_i=1$.
1
vote
0answers
67 views

A sum with binomial coefficients in the numerator and denominator.

I am struggling with a combinatorial sum as apart of a long statistical-mechanics derivation. I would appreciate any help. I seek the result of the following summation, for integer $\ell,n$, and ...
0
votes
1answer
29 views

Probability $\sum_{j=n+1}^{2n+1} {M \choose m+1}{M-m-1 \choose j-m-1}/{N \choose j} $

I have a prob. problem: A school has $N$ students in which $M$ students are leader (of each class in school), and $N>M$. There are $2n+1$ balls in the black box including $n+1$ blue balls and $n$ ...
0
votes
0answers
27 views

Find the closed-form of a series

Suppose that $x$ is positive number such that $x>0$. I just wonder is there existing a closed form of the series below $f(x)=\sum_{l=0}^{\infty}(2l+1)e^{-xl(l+1)}$. Is the well-known ...
1
vote
1answer
33 views

Transforming a power tower to a product

It is possible to write the product of a sequence of terms $a_i$ as a function of the sum of a sequence of functions of these terms: $$\prod_i a_i=f\left(\sum_i g(a_i)\right)$$ where $f=\exp$ and ...
-1
votes
4answers
111 views

How to calculate $k^0+k^1+k^2 + k^3+…+ k^{n-1}$ [duplicate]

How to simplify below expression or convert it to something simpler like $k^{n-1}$? $$ k^0+k^1+k^2 + k^3+...+ k^{n-1} $$
1
vote
1answer
51 views

Random walks with finite chance of escape

In a recent answer I gave a combinatorial interpretation for the sum $\sum_{n=1} \binom{2n}{n}\frac{4^{-n}}{n+1}=1$: namely, that it corresponded to the probability of all outcomes adding to $1$. A ...
1
vote
1answer
59 views

An upper bound and simplification for expression

I would like to find the upper bound (or simplification) of this expression: $$\sum_{j=1}^{n+1}\sum_{i=0}^{j-1} a^{j+i} {j+i \choose i}{n+1\choose j}{n \choose i}/{2n+1 \choose j+i}$$ where ...
1
vote
2answers
173 views

How find this sum $\sum_{i=0}^{2n}\binom{2n}{2i}\binom{2i}{i}y^{2i}$

Find the sum close form $$f(x)=\sum_{i=0}^{2n}\dfrac{\binom{2n}{2i}\binom{2i}{i}x^{2i}}{2^{2i}}$$ if we let $$\dfrac{x}{2}=y$$ then $$f(y)=\sum_{i=0}^{2n}\binom{2n}{2i}\binom{2i}{i}y^{2i}$$ ...
1
vote
0answers
32 views

Question about finite sums and integer recursions.

Let $n$ be a positive integer and let $g(n)$ be a given strictly increasing integer function such that $0<g(n)<n$ for all $n$. Also the sequence $ |g(n) - n|$ is unbounded as $n$ grows. Let ...