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

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2
votes
1answer
41 views

Why is the average of a sum equal the sum of the averages?

I came across this website showing the proof of the above question in the Expected Value section. However, I do not quite understand why the probability of $xy$ becomes the probability of $x$ (or $y$) ...
2
votes
2answers
53 views

Simplifying a Triple Summation

I have the summation: $$ \sum_{c=1}^{n-1} \sum_{k=c}^n \sum_j \frac{\rho(n,k)}{j!(k-c-j)!(c-j)!} $$ Where the sum $j$ goes from $0$ to $k-c$ if $k-c \leq c$, but if $k-c \geq c$ then the sum goes from ...
1
vote
1answer
23 views

How to write a covariance

Could somebody kindly explain how you can express something as a covariance? A paper I am reading contains this term: $$\sum_in_i(p_i-p)(b_a+\sum_t\sum_jr^t\frac {n_j} {n_i}b_c)$$ The author then ...
2
votes
1answer
30 views

Proving that the derivative of the sum of the first powers is $nP_{n-1}(x) + b_n$

Let $n \geq 0$ and consider $\sum\limits_{k=0}^x k^n$. It's a theorem that there is a polynomial $P_n(x)$ of degree $n+1$ such that $P_n(x) = \sum\limits_{k=0}^x k^n$. One way to determine these ...
4
votes
6answers
155 views

Why does the $\sum_{n=1}^x \frac {1}{n} \sim \mathrm {ln} x + \frac {1}{2}$

I was playing around with the Harmonic Series and I noticed that: $$\sum_{n=1}^x \frac {1}{n} \sim \mathrm {ln} x + \frac {1}{2}$$ I wanted to know if this is just some coincidence or if it is caused ...
1
vote
4answers
63 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
votes
3answers
81 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
votes
4answers
78 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
vote
1answer
39 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
vote
0answers
65 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
vote
2answers
112 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
votes
1answer
25 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
votes
3answers
153 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
votes
0answers
67 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 ...
0
votes
0answers
21 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
votes
2answers
68 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
votes
1answer
44 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
votes
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
votes
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
votes
3answers
38 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
votes
1answer
120 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
vote
1answer
46 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
votes
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
100 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
vote
0answers
41 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
votes
0answers
15 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
votes
2answers
68 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
votes
3answers
103 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
votes
1answer
53 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 + ...
16
votes
7answers
732 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
votes
0answers
43 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
vote
1answer
49 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
votes
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
votes
4answers
121 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 ...
-2
votes
2answers
44 views

Summation Problems [closed]

How did this particular equation come about? I haven't seen it before in the summation rules index on wikipedia: $$\sum\limits_{i=1}^{k+1} x_i =\left(\sum\limits_{i=1}^{k} x_i\right)+x_{k+1} $$ ...
-1
votes
7answers
200 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
votes
4answers
78 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
votes
1answer
100 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
votes
3answers
49 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
votes
3answers
72 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
vote
2answers
48 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 ...
0
votes
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 ...
1
vote
0answers
48 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
votes
1answer
223 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 ...
2
votes
1answer
56 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
votes
1answer
33 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
votes
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
votes
3answers
75 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
85 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 ...
1
vote
1answer
61 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, ...