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

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21 views

How to mathematically describe the number of Element x in a set

I am trying to formulate the following. I have a Set A={x, y, z}, I also have a Set B, C and D, which all are subsets of A. It is not exactly defined which elements are in B, C and D. I only want to ...
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0answers
8 views

Sum of values of a function with concurrent lines

Lets say I have a linear function as follows: $y=k*x_0$, where $k$ is NOT a constant. Values of $k$ can be between $0$ and $k_1$. In this example values of $k$ range from $k_2$ to $k_3$. My ...
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2answers
33 views

Proving Primness in a summation

I've been hitting my head against the wall for a little bit trying to figure out where to get started on proving (or disproving) the following: $\exists k \in \mathbb{Z} $ such that$ ...
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0answers
27 views

Finding the closed form of an infinite series

Consider the infinite series $$S=\sum^\infty_{j=0} \frac{\cos{\left((2j+1)\frac{\pi}{4}\right)}}{(2j+1)^2 \exp((2j+1)^2)}$$ Is there any way I could find the closed form expression? I have considered ...
3
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0answers
58 views

Can this summation be expressed differently?

Lets say I have a sum that states the following $$ \sum_{j=0}^{k-c} {k-c \choose j}\ln(a)^{k-c-j} \frac{d^j}{dx^j}[(x)_c] $$ where $(x)_c$ is the falling factorial such that $$ (x)_c = ...
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0answers
29 views

Approximate Sum of exponentials

I want to closely approximate the following sum into possibly a single term (not as infinite summation) $SUM= \exp^{(-a_1x_1-a_2x_2-a_3x_3)} - \exp^{(-v)}\exp^{(-b_1x_1-b_2x_2-b_3x_3)}$ here $a_i, ...
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1answer
48 views

Incomplete statement of commutativity of summation in Concrete Mathematics?

My text (p. 30, eq. 2.17) states the "commutative law" for sums as $$\sum\limits_{k\in K}{a_k} = \sum\limits_{p(k)\in K}{a_{p(k)}}$$ where $p$ is a permutation and $K$ is a set of integers. While ...
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2answers
56 views

n-th Derivative

It can be proven the for a function $h(x)=f(x)g(x)$, letting $f^{(k)}(x)=\frac{d^k}{dx^k}f(x)$ and $g^{(k)}(x)=\frac{d^k}{dx^k}g(x)$ then the n-th derivative, for n is an integer is: ...
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5answers
57 views

Exact closed form expression of $(2^0+…+2^n)+(2^1+…+2^{n+1})+…+(2^n+…+2^{2n})$

Exact closed form of this expression $(2^0+...+2^n)+(2^1+...+2^{n+1})+...+(2^n+...+2^{2n})$ I assume this means there is just one $2^0$ and one $2^{2n}$ and a double of all the terms in between?
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2answers
67 views

Intuition behind sum of multiplication arithmetic sequence

Maybe this is a stupid question but please guide and enlighten me patiently. I have just known something fact that quite shocking me. Let start from this simple fact $$\sum_{k=1}^n ...
2
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3answers
51 views

summation of $\sum^n_{k=0} (n-k)^2$

I'm trying to find the recurrence of $$ T(n) = T (n-1) + n^2$$ After following the steps, $$T (n) = T (n-1) + n^2 = T (n-2) + (n-1)^2 + n^2 $$ $$T (n) = T (n-2) + (n-1)^2 + n^2 = T(n-3) + (n-3)^2 ...
1
vote
1answer
69 views

Sums of consecutive odd integers, positive or negative

While supervising a student competition, my colleague and I ran across an interesting problem. Deobfuscated, it boils down to this Given a limit value $M$, which integers in the range ...
1
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1answer
18 views

Finite sum $\sum_{r,k} p_kP_r(x_k)f(x_k)P_r(x_m)=f(x_m)$

Let $x_0,\ldots,x_n\in\mathbb{R}$ be $n+1$ arbitrary real points and $p_0,...,p_n>0$ be positive real numbers. Let $P_0,P_1,\ldots,P_n$ be polinomials such that $$\sum_{k=0}^n ...
2
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2answers
55 views

On sums and identities

I am given the following problem set: (a) the Riemann $\zeta$-function for $s > 1$ is defined through the convergent sum: $$\zeta(s) := \sum_{n = 1}^{\infty} \frac{1}{n^s}$$ show the identity ...
3
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3answers
88 views

Find the exact closed from expression of $1^2 + 3^2 + 5^2 + · · · + (2n + 1)^ 2$ [duplicate]

I know the above expression equals to $\frac{n(2n−1)(2n+1)}{3}$, but how exactly can i come up with something from scratch?
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1answer
25 views

Conditions on $\lambda, \mu$ of $\sum_{n=0}^{n=\infty} \frac{\mu^{n}(2 \lambda + \mu)}{2^{n + 1} (\lambda + \mu)^{n + 1}} = 1$

What are the conditions on $\lambda \textrm{ and } \mu$ of $$ \sum_{n=0}^{n=\infty} \frac{\mu^{n}(2 \lambda + \mu)}{2^{n + 1} (\lambda + \mu)^{n + 1}} = 1 $$ where, $\lambda, \mu$ are positive ...
5
votes
1answer
65 views

How find this sum $\sum_{k=1}^{\infty}\frac{1}{1+a_{k}}$

Let $\{a_{n}\}$ be the sequence of real numbers defined by $a_{1}=3$ and for all $n\ge 1$, $$a_{n+1}=\dfrac{1}{2}(a^2_{n}+1)$$ Evaluate $$\sum_{k=1}^{\infty}\dfrac{1}{1+a_{k}}$$ My idea 1: ...
4
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1answer
70 views

How to show $\sum_{i=1}^{n-1} \frac{i(n-2)!}{(n-1-i)!n^{i+1}} \sim 1/n$

How can one compute the large $n$ asymptotics of $$\sum_{i=1}^{n-1} \frac{i(n-2)!}{(n-1-i)!n^{i+1}}\;?$$ My guess is that it is $1/n$ but I don't know how to show that.
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2answers
75 views

How to show that $\sum_{x=1}^\infty \prod_{i=1}^{x-1} (1-i/n) \sim \sqrt{\frac{\pi n}{2}}$?

How can one show that asymptotically $$\sum_{x=1}^\infty \prod_{i=1}^{x-1} \left(1-\frac{i}{n}\right) \sim \sqrt{\frac{\pi n}{2}} \; ?$$ A non rigorous argument is to say that for large $n$, ...
0
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1answer
281 views

Find all radial solutions of $\Delta u(\underline{x})=\frac{1}{(1+\parallel x\parallel^2)}$ on $\mathbb R^2\backslash\{0\}$

So far I've written $\Delta u(\underline x)=\Delta u(x_1,...,x_n)$ and I think that this equals this: $$\frac{\partial^2u}{\partial x_1^2}+...+\frac{\partial^2u}{\partial x_n^2}.$$ I also think that ...
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1answer
36 views

a_1,a_2,a_3…..a_n of natural numbers different then 0.

I understoof the logic behind it but have no idea how to put it into words. ex: n=3 ther are 4 n sum series: 1,1,1 1,2 2,1 3 of natural numbers different then 0, will be called n sum series if the ...
3
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1answer
75 views

Did Gauss find the formula for $1+2+3+\ldots+(n-2)+(n-1)+n$ in elementary school?

I heard Gauss's primary school teacher gave some busy-work to his class: to add all the numbers between 1 and 100 up. Gauss immediately wrote 5050. His teacher was shocked, so she told him to add up ...
0
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1answer
15 views

How do I find the value of this summation problem involving exponents?

I've looked around online quite a bit and still can't figure out exactly what to do here.
4
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3answers
56 views

Calculating sums

My maths teacher showed me something on how to calculate sums. Let's take an example: $$\sum_{k=1}^n k(k+1) = \sum_{k=1}^n k^2 + \sum_{k=1}^n k = \frac{n(n+1)(2n+1)}{6} + \frac{n(n+1)}{2} = ...
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0answers
21 views

On a summation manipulation

I have $R_t= \frac{1}{h} \sum_{j=0}^{h-1} E_tr_{t+j} + \theta_t$ where $E_tr_{t+j} = E[r_{t+j}| I_t]$. By subtracting $r_t$ from both sides and after some manipulations I should get: $$R_t - r_t= ...
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1answer
35 views

search for a theorem related to $\sum_{n=1}^{\infty}n^2\exp(-n^2)<\int_{\nu=1}^{\infty}\nu^2\exp(-\nu^2)$

I need to use the following inequality: $$\sum_{n=1}^{\infty}n^2\exp(-n^2)<\int_{\nu=1}^{\infty}\nu^2\exp(-\nu^2)d\nu\tag{1}$$ what is the name of such a theorem?
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2answers
58 views

What is an efficient equation to calculate the nth sum of the series -1 + 2 - 3 + .. + ( - 1)^n n?

Please tell me an efficient equation to calculate the nth sum of the series $$(-1) + 2 + (-3) + ... + (-1)^{n}\cdot n$$
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0answers
15 views

Making sense of a summation simplification

(This is all related to the action of Hecke operators on modular functions) Given is the sum \begin{equation} n^{2k-1} \sum_{\substack{a \geq1,\\ ad=n,\\0 \leq b < d}} d^{-2k} \sum_{ m \in ...
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0answers
33 views

Notation for a sum product

I'm struggling with notation for a sum product. Let $f:Z^+\rightarrow Z^+$. I am interested in a sum where each term is the product of functions whose sum of arguments equals $n$. For example if $n=3$ ...
2
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3answers
40 views

Limit of Sum to Infinity

Is $ \Sigma_{n=0}^\infty (\sqrt[3]{n^3+1} - n)$ convergent or divergent? For expressions of the form $\sqrt{n^2+1} - n$, I believe the common trick is to multiply by the "conjugate" ...
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4answers
103 views

Why is this nested sum formula true

I've been trying to get this sum: $\sum_{i}^{n} \sum_{j=0}^{n-i}j$ into a closed formula but couldn't really understand how to "unpack" that nested sum. It occured to me that the answer is: ...
3
votes
1answer
36 views

Find $\sum_{i=[1,\ldots,n], j=[1,\ldots,m]: i\neq j }\frac{1}{j}$

I have the following double $\sum_{i=[1,\ldots,n], j=[1,\ldots,m]: i\neq j }\frac{1}{j}$. If I ignore restriction $i \neq j$ then $$ \sum_{i=[1,\ldots,n], j=[1,\ldots,m] }\frac{1}{j}=n \sum_{ ...
2
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1answer
79 views

Prove the summation involving Stirling numbers of the first kind

I have been trying to prove or disprove this for 2 days now, but i don't even know where to begin. $$ 1 = \sum_{m=1}^{n} \sum_{k=1}^{n} \frac{x^{n-m}(-1)^{n-k-m} \left[\matrix ...
0
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1answer
194 views

By applying term-wise differentiation and integration find the sum of the series $\sum_{k=1}^{\infty}\frac{x^k}{k}$

I need to find the sum of the following series: $$\sum_{k=1}^{\infty}\frac{x^k}{k}$$ on the interval $x\in[a,b], -1<a<0<b<1$ using term-wise differentiation and integration. Can anyone ...
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0answers
32 views

Finite Sums and Riemann-Stieltjes integral

Show that every finite sum $\sum_{k=1}^{n}a_k$ can be written as a Riemann-Stieltjes integral. My thoughts As far as I understand step functions provide a bridging link between Riemann-Stieltjes ...
3
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0answers
90 views

Unusual binomial sum: $\sum_{d=k}^{n} {d \choose k} p^{d}(1-p)^{n-d}$

Does anyone know how to simplify the following sum? It's been giving me and everyone else I've showed it to quite a bit of trouble. I'm quite confident that this should simplify, but I just can't seem ...
2
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1answer
97 views

Finding a more explicit way to express a coefficient in this summation

I am sorry about the vague title, i really don't know how else to ask the question. So i have came up with the following: $$ 1 = \sum_{k=0}^{n} \sum_{v=0}^{n} \frac{x^{n-k+1} T_{n+1,k+1} ...
3
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4answers
67 views

Algebric proof for the identity $n(n-1)2^{n-2}=\sum_{k=1}^n {k(k-1) {n \choose k}}$

Prove the identity: $$n(n-1)2^{n-2}=\sum_{k=1}^n {k(k-1) {n \choose k}}$$ I tried using the binomial coefficients identity $2^n = \sum_{k=1}^n {n \choose k}$ but got stuck along the way.
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0answers
53 views

Prove $\forall n\in\mathbb{N}, \exists m\in\mathbb{N}; n=\pm1^2\pm2^2\pm\cdots\pm m^2.$

And we choose the positive and negative signs in a way that the equation becomes true. I think it can be proved with mathematical induction. So here's how I begin: For $n=1$, $1=+1^2$ which is true. ...
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3answers
61 views

How to prove $ 1^3+2^3+… +n^3 = (1 + 2+ \dots +n)^2 $ by induction? [duplicate]

I need to prove that for each natural n: $$ 1^3+2^3+... +n^3 = (1 + 2+ ...+n)^2 $$ How do I do that? how do I know whether I should choose strong induction or simple induction?
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1answer
58 views

Combinational meaning of $\sum_{k=r}^{n} {k \choose r }={{n+1} \choose r+1}$

What's the combinational interpretation of the identity $\sum_{k=r}^{n} {k \choose r }={{n+1} \choose r+1}$?
5
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2answers
142 views

Closed form for a zeta series

It is not that diffcult to derive \begin{align} \sum^\infty_{k=2}\frac{(-1)^{k-1}\zeta(k)}{k2^k}=&-\frac{\gamma}{2}+\ln\left(\frac{2}{\sqrt{\pi}}\right)\tag{1}\\ ...
0
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1answer
53 views

Prove $\sum_{k=0}^{n-1}\frac{n(-1)^{n-k+1}}{n-k}{n-1 \choose k} = 1$

To finish a proof i have been working on i must prove the following: $$ \sum_{k=0}^{n-1}\frac{n(-1)^{n-k+1}}{n-k}{n-1 \choose k} = 1 $$ I have checked that it does work empirically, but of course ...
4
votes
1answer
89 views

Why are infinite sums so much harder to calculate than the associated infinite integral?

It seems that with continuous functions, we have in calculus an apparatus for "short cutting" an infinite sum. However, when we move to the discrete case, it seems that we don't have the equivalent ...
0
votes
1answer
44 views

Find a formula for the sum of the $n$ terms of the sequence (Summations & Sequences) [duplicate]

Find a formula for the sum of the $n$ terms of the sequence: $1, 1 + 2, 1 + 2 + 2^{2}, 1 + 2 + 2^{2} + 2^{3}, ...$ My Approach: When $n$ increases the sequence increases by $2^{n}$ for every $n$. ...
2
votes
2answers
28 views

Solve easy sums with Binomial Coefficient

How do we get to the following results: $$\sum_{i=0}^n 2^{-i} {n \choose i} = \left(\frac{3}{2}\right)^n$$ and $$\sum_{i=0}^n 2^{-3i} {n \choose i} = \left(\frac{9}{8}\right)^n.$$ I guess I could ...
0
votes
1answer
24 views

Einstein summation convention differential

I'm just learning this convention, and at times I'm a little lost, like now. I have to calculate the following, knowing that $a_{ij}$ are constants such that $a_{ij}=a_{ji}$: $$ ...
0
votes
1answer
50 views

Proving Euler Summation by Parts Without Using Integration by Parts

Assume $f$ has continuous derivative $f'$ on [a,b]. Prove the following summation formula, without using partial integration: \begin{equation} \sum_{a< x \le ...
4
votes
3answers
53 views

How to solve $ \sum_{k=1}^\infty \frac{1}{(2k-1)(2k+1)}$

Can you help me please with this sum? $$\sum_{k=1}^{\infty} \frac{1}{(2k-1)(2k+1)}$$ I have no idea how to solve it. Result is $\frac{1}{2}$. Thank you
3
votes
0answers
22 views

Properties of digit functions for numbers in $[0,1]$

Consider a function $g(n): \mathbb N \to \{0,1,2,3,4,5,6,7,8,9\}$, ie. $g$ maps the natural numbers to natural numbers between $0$ and $9$. Then, no matter what $g(n), \ n\in \mathbb N$ is, the sum ...