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

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4answers
66 views

Prove that $\sum_{i=1}^{i=n} \frac{1}{i(n+1-i)} \le1$

$$f(n)=\sum_{i=1}^{i=n} \dfrac{1}{i(n+1-i)} \le 1$$ For example, we have $f(3)=\dfrac{1}{1\cdot3}+\dfrac{1}{2\cdot2}+\dfrac{1}{3\cdot1}=\dfrac{11}{12}\lt 1$ If true, it can be used to prove: ...
4
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3answers
83 views

Proof of $\sum^{2N}_{n=1} \frac{(-1)^{n-1}}{n} = \sum^{N}_{n=1} \frac{1}{N+n}$

The title pretty much summarizes my question. I am trying to prove the following: $$\displaystyle \forall N \in \mathbb{N}: \sum^{2N}_{n=1} \frac{(-1)^{n-1}}{n} = \sum^{N}_{n=1} \frac{1}{N+n}.$$ I ...
0
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4answers
79 views

Absolute convergence of $\sum_{n=1}^{\infty} \frac{\sin(2n)}{\sqrt{n}}$

I have to discuss the conditional and absolute convergence of the series: $$\sum_{n=1}^{\infty} \frac{\sin(2n)}{\sqrt{n}}$$ I believe such a series is conditionally convergent but not absolutely ...
1
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1answer
70 views

Interchange finite and infinite sum

Under which condition is it valid to interchange a finite and an infinite sum? We have used $$\sum_{x \in I} \sum_{y=0}^{\infty} f_{x,y}= \sum_{y=0}^{\infty} \sum_{x \in I} f_{x,y}$$ for a finite ...
1
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0answers
72 views

Stirling number Combinatorics. Summation .

$$ \sum_{k=0}^n \left\{ {n\atop k} \right\} *(x)_k = x^n $$ is well known . What if the k-th term of LHS summation is divided by $q^k$ where $q$ is some positive constant, What about $$ ...
1
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2answers
115 views

Alternating sum of product of Fibonacci numbers

Suppose that $\{F_n\}$ is the sequence of Fibonacci numbers. There is a well-known result that $$\sum_{i=1}^nF_i^2 F_{i+1}=\frac{1}{2}F_nF_{n+1}F_{n+2}.$$ This is easy to prove by induction. I was ...
0
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1answer
27 views

Sum of a product of four Kronecker Deltas

The Kronecker delta has the following property: $$\sum_{k} \delta_{ik}\delta_{kj} = \delta_{ij}. $$ Does anyone know whether the following formula is correct? $$\sum_{i=1}^N ...
10
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3answers
157 views

How can I get the exact value of this infinite series?

I want to compute the exact value of this infinite series $$\sum_{n=2}^\infty\arcsin{\left(\dfrac{2}{\sqrt{n(n+1)}(\sqrt{n}+\sqrt{n-1})}\right)}$$ By comparison test, we can get the series is ...
5
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0answers
68 views

Infinite Sums which turn out to be Riemann Integrals

I'm looking for examples of infinite series which look hard to evaluate at first, but become very simple when viewed as a Riemann integral. An example would be $$\frac{1}{n+1}+\frac{1}{n+2}+ \ldots ...
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0answers
26 views

Pairing function output that can be summed

Is there a pairing function that can take in a set of natural numbers N with a known set length and output a single natural number ...
0
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2answers
69 views

What is sum: $\sum\limits_{m,n\geq1}\frac{1}{(1+mn)^2}$?

What is the sum $$\sum\limits_{m,n\geq1}\frac{1}{(1+mn)^2}.$$
-2
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1answer
52 views

Sum of all the numbers with the given numbers repeated

How to find the sum of all the numbers that can be formed using the digits 4,5,5,6,6,6 (This includes 4,5,6,45,46,54,55,....,666554). I knew that the answer is 39345806. I just need to know the method ...
3
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1answer
45 views

How shall I calculate $\sum\limits_{d\nmid n}\mu(d)$

Today when I was studying Apostol's Analytical Number theory, I came to know about the formula $\sum\limits_{d|n}\mu(d)=1$ if $n=1$ and $0$ otherwise. I understood the technique and then using the ...
0
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0answers
33 views

A complicated summation of binomial coefficients

I am trying to evaluate this sum. I think closed form of this sum is not possible, but there might be some bound or approximate result. So far I was unable to find any approximation. Any help will be ...
4
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3answers
39 views

Interchanging order of summation mechanically

How can I interchange order of summation mechanically, without thinking? For instance, I had to interchange the sums below (assume $i$ is a constant where $i\gt 0$). $$\sum_{n\ge 1}\sum_{i\lt k \lt ...
4
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2answers
127 views

A summation involving multinomial coefficient

We need to find out $$\sum {\binom{N}{a_1,a_2,a_3...a_B} a_1^{\alpha}a_2^{\alpha}...a_C^{\alpha} }$$ $$a_1+a_2...a_B=N, \alpha>0 ,0\lt C \le B$$ All are nonnegative integers. We need to sum ...
1
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1answer
47 views

Trying to understand a power series example from Advanced Calculus by Taylor

Example 2 from 21.1 in the book, Find an expansion in powers of $x$ of the function $$ f(x) = \int_{0}^{1} \frac{1-e^{-tx}}{t}dt $$ and use it to calculate $f(1/2)$ approximately. I ...
0
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1answer
64 views

Summation of $\frac{1}{k^2 - k}$ from $k=2$ to $\infty$. [duplicate]

I couldn't get an idea how to get this summation?Can you help me please!!
4
votes
2answers
101 views

Progression of the reciprocal of squares $ \lt \frac{1}{4}$

$$\frac{1}{9}+\frac{1}{16}+\frac{1}{25}+\frac{1}{36}+\frac{1}{121}\cdots \lt\frac{1}{4}$$ This is an interesting summation in which the addition of the next term must make the sum $\lt\frac{1}{4}$. ...
1
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0answers
43 views

Proving that $P_{k+1}(x) = 1 + \sum\limits_{j=0}^{k+1} \binom{k+1}{j}P_j (x-1)$

I'm struggling a little bit with this proof from Smoryński's Logical Number Theory. He has already proven that, if $n \geq 0$, then there's a polynomial $P_n(x) = \sum\limits_{k=1}^x k^n$. The idea ...
0
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0answers
19 views

Bernstein polynomial

I need some help in the following task. The i-th Bernstein polynomial of degree n on the interval [a,b] is $B_{i}^{n}(x;a,b) = (b-a)^{-n}\binom{n}{i}(b-x)^{n-i}(x-a)^{i}$ Show: The control points of ...
0
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2answers
85 views

What could be the mathematical equation of the given signal?

We know that Fourier series for periodic signal $y(t)$ is given by $$ y(t) = \sum\limits_{m=0}^{+\infty} a_m \cos(w_m t) + \sum\limits_{m=0}^{+\infty}b_m \sin(w_m t). \quad (2)$$ Now,I want to find ...
0
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3answers
105 views

How to reach $\dfrac{(n-1)n(2n-1)}{6n^3}$ [duplicate]

I am trying to refresh my Maths after a lot of years without studying them, and I am finding a lot of difficulties (which is actually nice). So, my question: I don't understand the next equality. How ...
0
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1answer
56 views

How to generalize the summation [closed]

For some work of mine, I came out with following terms for the value $m$, where $m$ is even. For $m=6$ : ${}\quad 1 + 6 + \{5+4+3+2+1\} + \{4+3+2+1\} = 32$ For $m=8$ : ${}\quad 1 + 8 + ...
15
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7answers
750 views

Summation Theorem how to get formula for exponent greater than 3

I'm studying in the summer for calculus 2 in the fall and I'm reading about summation. I'm given these formulas: \begin{align*} \sum_{i=1}^n 1 &= n, \\ \sum_{i=1}^n i &= \frac{n(n+1)}{2},\\ ...
3
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0answers
44 views

Value of double sum of powers of fractions between 0 and 1

Is there any way to find closed form for the sum (where k is positive integer) $$S = \sum_{i = 1}^{n}\sum_{j = 0}^{i} \left( \frac{j}{i} \right) ^ k$$ Using Faulhaber's formula I got $$S = ...
1
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1answer
58 views

Geometry formulas, how to show identities.

Given $d$ is integer: How do I show: $$\frac{1}{(e^{\frac{2i\pi p}{d}}-1)}=\frac{-i}{2\tan(\frac{\pi p}{d})}-\frac{1}{2}$$ How do I rewrite and show, for $k$ is an integer: $$ ...
0
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1answer
20 views

Convergence and Irrationality of $\frac{H_{(n,-n)}}{(n+1)^n}$ as $n$ approaches infinity

We define $H_{(a,b)}$ as the $a^{th}$ harmonic number of class $b$. In other words, $$H_{(a,b)}=\sum_{k=1}^a \frac{1}{k^b}$$ More information about generalized harmonic numbers can be found here. Let ...
6
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4answers
129 views

Finding $\sum\limits_{k=0}^n k^2$ using summation by parts

Sorry to bother you guys again, but I still have some doubts. I do think I'm making some progress, though. So, again, the formula that I'm using for summation by parts is $\sum\limits_{k=o}^n ...
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2answers
57 views

Summation Problems [closed]

How did this particular equation come about? I haven't seen it before in the summation rules index on wikipedia: $$\sum\limits_{i=1}^{k+1} x_i =\left(\sum\limits_{i=1}^{k} x_i\right)+x_{k+1} $$ ...
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7answers
205 views

Sum: $1-2+3-4+5-6+…$

If we forget all the rules about infinte sums what am I doing wrong? $$1-2+3-4+5-6+...=\sum_{n=1}^{\infty} n(-1)^{n+1}$$ (with Grandi's series) $$1,1+(-2)=-1,1+(-2)+3=2,1+(-2)+3+(-4)=-2,...$$ we ...
5
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4answers
86 views

Interesting summation question: If $a$ and $b$ are the roots of $x^2+x+1$, then what is the below expression equal to?

Question: If $a$ and $b$ are the roots of $x^2+x+1$, then what is the below expression equal to? $$\sum_{n=1}^{1729} \left[(-1)^n\cdot V(n)\right]$$ Where $$V(n)=a^n+b^n$$ My effort: I think I ...
5
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1answer
102 views

Finding $\sum\limits_{k=0}^n k$ using summation by parts

This is another exercise from Smoryński's Logical Number Theory; not being a mathematician, I'm a bit new to this finite difference stuff, so, please, bear with me! In a previous exercise, Smoryński ...
2
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3answers
50 views

Continuity of function consisting of an infinite series.

Let $f(x) , 0\leq x\leq 1$ be defined by, $$f(x)=\sum_{n=1}^{\infty}\frac{1}{(x+n)^2}$$. Show that $f$ is continuous on $[0,1]$ and that, $$\int_0^1f(x)dx=1$$. I have never dealt ...
2
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3answers
74 views

Calculating $\sum_{k=0}^{n}\sin(k\theta)$ [duplicate]

I'm given the task of calculating the sum $\sum_{i=0}^{n}\sin(i\theta)$. So far, I've tried converting each $\sin(i\theta)$ in the sum into its taylor series form to get: ...
1
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2answers
50 views

Is $\sum_{n=1}^\infty a_n\sin(nx)$ converges on $[\varepsilon, 2\pi-\varepsilon]$?

Let $a_n$, a sequence monotonically decreasing to $0$. Consider $$\sum_{n=1}^\infty a_n\sin(nx)$$ Is the series converges uniformly on $[\varepsilon, 2\pi-\varepsilon]$? ($\varepsilon ...
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1answer
42 views

Multiplication of 2 sums that equal another multiplication of 2 sums

I have been trying to prove a formula of mine and i come across something very interesting, well to me it is. If the formula is correct, it states that: $$ \left(\sum_{m=0}^{k-c} {k-c \choose m}{ms_1 ...
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0answers
52 views

How to Evaluate this Summation to Find a Closed Form

While taking the incomplete Bell Polynomil of $x^a$ i found out that: $$ B_{n,k}^{x^a}(x) = x^{ak-n} \sum_{m=0}^k \frac{(am)!(-1)^{k-m}}{m!(k-m)!(am-n)!} $$ Now, what i am wondering is, what is the ...
5
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1answer
253 views

Generalized Sophomore's dream. Question about originality

A few months ago I derived a beautiful fact: $$ \sum_{n=k+1}^\infty n^{k-n}=\int_{0}^{1} t^{k-t}dt~~~(*) $$ for every natural $k$. Generally: $$ \sum_{n=1}^\infty ...
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1answer
57 views

How do I calculate these sum-of-sum expressions in terms of the generalized harmonic number?

I know that $$\sum_{m=2}^k\sum_{n=1}^{m-1}(nm)^{-s}=\frac 12((H_k^s)^2-H_k^{(2s)})$$ and $H_k^s=\sum_{n=1}^kn^{-s}$ But, how would I go about finding identities in terms of the harmonic number like ...
0
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1answer
35 views

Formula for $\sum_{i = 1}^n k^n$ [duplicate]

I know from my calculator the answer is $\sum_{i = 1}^n k^n$ = $\frac{k^{n+1}-k}{k - 1}$. I'd just like help understanding why.
4
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2answers
82 views

Showing $\sum_{n=1}^\infty \sin x \sin nx$ is uniformly bounded

I need to show that for every $x$: $$\sum_{n=1}^\infty \sin x \sin nx \lt M$$ So the first thing came into my mind is applying a well-known trigonometric identity: $$\sum_{n=1}^\infty \sin x \sin nx ...
2
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3answers
76 views

value of an $\sum_3^\infty\frac{3n-4}{(n-2)(n-1)n}$

I ran into this sum $$\sum_{n=3}^{\infty} \frac{3n-4}{n(n-1)(n-2)}$$ I tried to derive it from a standard sequence using integration and derivatives, but couldn't find a proper function to describe ...
4
votes
1answer
88 views

Find the remainder when the sum is divided by $1000$

Find $S \pmod{1000}$ given: $$S = \sum_{n=0}^{2015} n! + n^3 - n^2 + n - 1$$ $$S_0 = 0! + 0 - 0 + 0 -1 = 0$$ $$S_1 = 1! + 1 - 1 + 1 - 1 = 1$$ $$S_2 = 2! + 8 - 4 + 2 - 1 = 7$$ This isn't ...
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1answer
64 views

How to Split a Sequence of Numbers Into Four (Relatively) Equal Summations

How would I go about splitting a sequence of numbers into four equal (as equal as possible) summations? Say I have a sequence of 26 integers like so: 16, 4, 17, 10, 15, 4, 4, 6, 7, 14, 9, 17, ...
0
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1answer
37 views

How to evaluate this combination of sums and integrals?

I am reading a book on PDEs, and I am near the beginning where the author is talking about the heat equation and, specifically, solving the non-homogenous equation $u_t={\alpha}^2u_{xx}+f(x,t).$ The ...
1
vote
2answers
124 views

Sum of trigonometric infinite series

I am trying to prove that for any $x\geq 1$ we have: $$ \sum_{m=1}^{\infty} \frac{\sin\frac{(2m-1)\pi}{x}}{\left(\frac{(2m-1)\pi}{x}\right)^3} = \frac{x}{8}(x-1). $$ Could I have some help please? I ...
3
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2answers
32 views

Finite double sum: Improve index transformation

In order to prove a rather complicated binomial identity a small part of it implies a transformation of a double sum. The double sum and its transformation have the following shape: ...
1
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3answers
106 views

Sum of $\sum\limits_{x=-\infty}^{\infty}x^{\operatorname{sign}(x)}$

Both the sum of $1+2+3+4+\cdots$ and the sum of $\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\cdots$ diverge. If both are paired together in one function, as seen above, can they amount to a ...
1
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2answers
71 views

How to prove that $\lim\limits_{n \to \infty} \sum\limits_{k = 1}^{n} {\sqrt k \over n^{1.5}} = {2 \over 3}$

I am currently trying to prove: $\lim\limits_{n \to \infty} \sum\limits_{k = 1}^{n} {\sqrt k \over n^{1.5}} = {2 \over 3}$ I can easily squeeze the series between 0 and 1. I don't know many handy ...