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

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4
votes
4answers
781 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
47 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
26 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
54 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
27 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
78 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
32 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
30 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
40 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
116 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
59 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
61 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
179 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
33 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 ...
0
votes
2answers
66 views

Simplify the expression of binom

Any one knows how to simplify this expression or finding upper bound of this expression: $$\sum_{j=1}^{n+1}\sum_{i=0}^{j-1} a^{j+i} {j+i \choose i}$$ where $0<a<1$ is constant. Thanks a lot.
0
votes
0answers
29 views

Simplifying a combinatorial sum

Show that \begin{align} y\sum\limits_{i=1}^dx^iz^i\sum\limits_{j=1}^iq^{i-j}G_d(x,y,q\mid j) = y\sum\limits_{i=1}^d(x^iz^i+\cdots+x^dz^dq^{d-i})G_d(x,y,q\mid i) \end{align} where \begin{align} ...
0
votes
2answers
76 views

Compound interest with a compounding interest rate

I have an investment which pays 3% interest (r) annually but it also increases the interest rate every year by 5% (g). I re-invest all interest payments at the start of each year. How many years (t) ...
7
votes
3answers
633 views

How do you calculate this limit $\lim_{n\to\infty}\sum_{k=1}^{n} \frac{k}{n^2+k^2}$?

How to find the value of $\lim_{n\to\infty}S(n)$, where $S(n)$ is given by $$S(n)=\displaystyle\sum_{k=1}^{n} \dfrac{k}{n^2+k^2}$$ Wolfram alpha is unable to calculate it. This is a question from a ...
1
vote
1answer
33 views

Extracting the coefficient of $x^n$ from a fraction

I need help extracting the coefficient of $x^n$ from a $\frac{1-x}{1-2x}$. So far I have that \begin{align} \frac{1-x}{1-2x} &= \frac{1}{1-2x} - x\frac{1}{1-2x}\\ &= \sum\limits_{k=0}(2x)^k ...
1
vote
1answer
34 views

Upper bound of $\sum\limits_{j=1}^{n+1} \sum\limits_{i=0}^{j-1}{n+1 \choose j}{n \choose i}$

I would like to find max (or sup.) of the sum: $$S=\sum\limits_{j=1}^{n+1} \sum\limits_{i=0}^{j-1}{n+1 \choose j}{n \choose i}.$$ I found $S\le \frac{1}{\sqrt{\pi n}}.2(n+1).4^n$ but It seems it's ...
1
vote
1answer
79 views

Help on a tough summation from Rudin?

I'm having a tough time deriving (4) from the bracketed expression in (3) shown in the photo. I've been futzing with partial sums of geometric series and binomial expansions for a while now with no ...
2
votes
0answers
31 views

A question about the differentiability of two Weyl sums

Consider the following functions, associated with certain trigonometrical sums: $$ f_{\alpha,\beta}(x) = \sum_{n=1}^{+\infty}\frac{\cos(n^{\alpha+\beta}x)}{n^{\alpha}},\qquad g_{\alpha,\beta}(x) = ...
0
votes
0answers
29 views

Finding alternating series for Power series

Given data and conditions I have a power series, $PS(x) = \sum_{n=0}^\infty R_nx^n$. I have a infinite GP,something like G(x) = $\sum_{k=0}^\infty ax^k = \frac{a}{1-x} $ . Never take G(x),such ...
9
votes
4answers
193 views

A closed form of $\sum_{k=0}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right)$

I am looking for a closed form of the following series \begin{equation} \mathcal{I}=\sum_{k=1}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right) \end{equation} I have no idea how to ...
0
votes
1answer
27 views

Rank of a simple matrix series

Problem Specifications and Given conditions I have a matrix $L$ with rank 3 and dimension $ 3 \times 3$. $L = K_0+\sum_{n=1}^{\infty}K_i $ . Rank of $K_0$ is 3 and rank of L is also 3. Rank of ...
5
votes
3answers
101 views

Ordered partitions of an integer (with a twist)

I would like to know how to prove (preferably algebraically) that $P_1(2,n)=F_{2n+1}$, where $P_1(2,n)$ is what I define to be the number of ordered partitions of an integer, where the number $1$ has ...
1
vote
2answers
58 views

Matrix Inversion Test ( Sum of Matrix series)

Friends,I have a set of matrices of dimension $3\times3$ called $A_i$. , Following are the given conditions a) each $A_i$ is non invertible except $A_0$ because their determinant is zero. b) ...
1
vote
0answers
30 views

a question on sum of q_binomials

I was trying to calculate something and at some point I get the following sum: \begin{equation} \sum_{t=0,t \text{ even}}^{s}{s+3n \brack s-t}\sum_{i = 0}^{t/2}q^{2i^2}{t/2+2n-i \brack t/2-i}{n ...
0
votes
2answers
30 views

Question about partial sum

I'm confused on the partial sums formula Why is $$\sum_{i=m+1}^\infty \frac{2}{3^i}=\frac{1}{3^m},$$ if $$\sum\limits_{k = 0}^\infty\frac{2}{3^k} = 3.$$
1
vote
1answer
36 views

Question about written proof for geometric summation

Suppose $\alpha$ $\ne$ $\beta$ $\in \{0, 2\}^\mathbb{N}$ Prove that $$\sum\limits_{k = 0}^\infty\frac{\alpha(k)}{3^k} \ne \sum\limits_{k = 0}^\infty\frac{\beta(k)}{3^k}. $$ This is the written proof ...
1
vote
1answer
55 views

Sum of nth powers and generalized polynomial sum

So this is a 2-part question (both parts I believe are closely related): How exactly does on express the sum $$\sum_{i=0}^{k}{i^n} = Q(n,k)$$ in a closed form For arbitrary positive integers ...
15
votes
2answers
569 views

Intuitive ways to get formula of cubic sum

Is there an intuitive way to get cubic sum? From this post: combination of quadratic and cubic series and Wikipedia: Faulhaber formula, I get $$1^3 + 2^3 + \dots + n^3 = \frac{n^2(n+1)^2}{4}$$ I think ...
3
votes
1answer
102 views

find a $B_{n,j}$ such that $|A_{n,j}-L_j| \leq B_{n,j}$ $\forall n,j$ and $\sum_{j=0}^{\infty}B_{n,j}$ converges

We have $A_{n,j}= 3(-1)^j2^{n-j+1}\frac{(2(n-j)-4)!}{(n-j)!(n-j-2)!}\binom{j+2}{2}\frac{n^\frac{5}{2}}{8^n}$ and $L_j=(-\frac{1}{8})^j\binom{j+2}{2}\frac{3}{8\sqrt{\pi}}$ So I know $\lim_{n \to ...
19
votes
4answers
550 views

Calculate $\frac{1}{5^1}+\frac{3}{5^3}+\frac{5}{5^5}+\frac{7}{5^7}+\frac{9}{5^9}+\cdots$

I'm an eight-grader and I need help to answer this math problem. Problem: Calculate $$\frac{1}{5^1}+\frac{3}{5^3}+\frac{5}{5^5}+\frac{7}{5^7}+\frac{9}{5^9}+\cdots$$ This one is very hard for ...
2
votes
1answer
121 views

Combination of quadratic and cubic series

I'm an eight-grader and I need help to answer this math problem (homework). Problem: Calculate $$\frac{1^2+2^2+3^2+4^2+...+1000^2}{1^3+2^3+3^3+4^3+...+1000^3}$$ Attempt: I know how to ...
2
votes
1answer
51 views

Question about proof with geometric sums

I am confused on how to write proofs for geometric sums. I think that using the well ordering principle to find the least n $\in$ $\mathbb{N}$ with $\alpha(n)$ $\ne$ $\beta(n)$ would be a good ...
1
vote
2answers
47 views

Question about proof with sums

I am confused on how to write a formal proof for sum notations. How would I write a formal proof for this example? Prove that $$\sum\limits_{k = 0}^\infty\frac{2}{3^k} = 3.$$ Prove that for any ...
3
votes
1answer
76 views

Upper bound of $S=\sum_{k=1}^{P}k!\binom{P}{k}\binom{Q}{k}$

EDIT: How can I find a good upper bound to this quantity ? $$S_{n,m}=\sum_{k=1}^{P}k!\binom{P}{k}\binom{Q}{k}$$ where $P=\min\{m,n\}$ et $Q=\max\{m,n\}$.
0
votes
0answers
18 views

Does a piece wise function inside a sum make sense (fractional bias)?

I'm looking for a good way to express the calculation for the fractional bias. It is used to compare simulated values ($s$) with observed values ($o$) and pretty simple to calculate: build the ...
0
votes
1answer
31 views

How to simplify an expression where the summation is over all subsets of a given set?

I want to simplify this expression: $\sum\limits_{\emptyset \ne I \subseteq \mathbb{N}_k}(-1)^{|I|-1}|A_I|+|A_{k+1}|-\sum\limits_{\emptyset \ne I \subseteq \mathbb{N}_k}(-1)^{|I|-1}|A_I \cap ...
0
votes
1answer
50 views

How to calculate the sum of combinatorial numbers

For my work on an almost completely unrelated field I came across the following formula. I know that I have learned that all in high school. But since this is more than 15 years ago in which I never ...
3
votes
1answer
49 views

Measuring sums of complex alternating series

Suppose we have functions $$f(x) = \sqrt{x}, \space g(f) = \frac{df}{dx}+\frac{d^2f}{dx^2}+\frac{d^3f}{dx^3}\space ...$$ Applying function f(x) to g(f) we get: $$g(f(x))=\frac{1}{2}x^{-\frac{1}{2}} - ...
7
votes
6answers
253 views

Finding $\binom{n}{0} + \binom{n}{3} + \binom{n}{6} + \ldots $

Help me to simplify:$$\binom{n}{0} + \binom{n}{3} + \binom{n}{6} + \ldots $$ I got a hunch that it will depend on whether $n$ is a multiple of $6$ and equals to $\frac{2^n+2}{3}$ when $n$ is a ...
2
votes
2answers
57 views

is there a generating function for $H_{2n}$?

I have been wondering if anyone knows if there is a generating function for harmonic series of the form $H_{2n}$?. That is, we are familiar with ...
0
votes
1answer
28 views

Easy question regarding this proof

I do not understand a small step in a proof I'm reading at the moment. Why are the following things equal? $$\sum_{k=1}^{n} \frac{1}{2k-1} - \frac{1}{2} \sum_{k=1}^{n} \frac{1}{k} = \sum_{k=1}^{2n} ...
4
votes
2answers
39 views

$\sum_j e^{i\phi_j}$ vs $\sum_j e^{ip\phi_j}$

Let $\phi_j$ be a collection of angles. If $p$ is a positive integer, how is the sum $\sum_je^{i\phi_j}$ related to $\sum_je^{ip\phi_j}$?
14
votes
3answers
455 views

Equality of sums with fractional parts of the form $\sum_{k=1}^{n}k\{\frac{mk}{n}\}$

I recently encountered the following equality ($\{\}$ denotes fractional part): $$\sum_{k=1}^{65}k\left\{\frac{8k}{65}\right\}=\sum_{k=1}^{65}k\left\{\frac{18k}{65}\right\}$$ and found it very ...
6
votes
1answer
259 views

Closed form $\sum_{n=2}^{\infty} \frac{1}{\ln^n{n}}$ and $\sum_{n=2}^{\infty} \frac{n}{\ln^n{n}}$

Apologies if this has been asked before, but I was playing around with Wolfram Alpha and got approximations but not closed forms for $$\sum_{n=2}^{\infty} \frac{1}{\ln^n{n}} \approx 3.2426094109 $$ ...
1
vote
2answers
105 views

How to find $\sum_{r=1}^{n} r^2\cos {(r\theta)}$

How do you find the sum $$S(\theta)=\displaystyle\sum_{r=1}^{n} r^2\cos {(r\theta)}$$? I observed that if $f(\theta)$ is $\sum\cos {(r\theta)}$, then $S(\theta)=-f''(\theta)$. Help will be ...