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
1k views

How to show that this binomial sum satisfies the Fibonacci relation?

The binomial sum $$s_n=\binom{n+1}{0}+\binom{n}{1}+\binom{n-1}{2}+\cdots$$ satisfies the Fibonacci relation. I failed to prove that $\binom{n-k+1}{k}=\binom{n-k}{k}+\binom{n-k-1}{k}$... Any ...
2
votes
2answers
193 views

Double Summation: Need help to handle $ i \neq j $ : $ \sum_{i=0 \to 7,\ j=1 \to 8,\ i\neq j} (8i + j) $

[Q1]. Can I ? ( write the same summation as ) : $$ \sum_{i=0, i \neq j}^7 \sum_{j=1}^8 (8i + j) \tag{1}$$ I tried to solve the following Summation as follows: Let i = m-1 then, $ \sum_{i=0,\ i ...
-1
votes
1answer
86 views

Use Mathematical Induction to prove that $\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} +…+\frac{1}{n(n+1)}=1-\frac{1}{n+1}$

Use Mathematical induction to prove that for all integers, $n$ is greater than or equal to $1$. I am confused on what to do after I do the the basis step that is using $n$ as $1$. $$\frac{1}{1 \cdot ...
28
votes
1answer
514 views

Geometric interpretation for sum of fourth powers

Summing the first $n$ first powers of natural numbers: $$\sum_{k=1}^nk=\frac12n(n+1)$$ and there is a geometric proof involving two copies of a 2D representation of $(1+2+\cdots+n)$ that form a ...
10
votes
2answers
244 views

Is $\sum_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}=\frac{m^2-1}{3}$ true for $m\in\mathbb N$?

Question : Is the following true for any $m\in\mathbb N$? $$\begin{align}\sum_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}=\frac{m^2-1}{3}\qquad(\star)\end{align}$$ Motivation : I reached ...
21
votes
5answers
1k views

Sum of the form $r+r^2+r^4+\dots+r^{2^k} = \sum_{i=1}^k r^{2^k}$

I am wondering if there exists any formula for the following power series : $$S = r + r^2 + r^4 + r^8 + r^{16} + r^{32} + ...... + r^{2^k}$$ Is there any way to calculate the sum of above series (if ...
8
votes
4answers
380 views

Sum : $\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)^3}$

Prove that : $$\sum_{k=0}^{\infty} \frac{(-1)^{k}}{(2k+1)^3}=\frac{\pi^3}{32}.$$ I think this is known (see here), I appreciate any hint or link for the solution (or the full solution).
10
votes
2answers
233 views

Show that $\displaystyle\sum_{k=0}^n\binom{2n}{2k}^{\!2}-\sum_{k=0}^{n-1}\binom{2n}{2k+1}^{\!2}=(-1)^n\binom{2n}{n}$

How can I prove this $$ \sum_{k=0}^n\binom{2n}{2k}^2-\sum_{k=0}^{n-1}\binom{2n}{2k+1}^2=(-1)^n\binom{2n}{n}? $$ Maybe can we expand $$ f(x)=(1+x)^{2n}? $$ Thank you.
20
votes
4answers
2k views

Identity involving Euler's totient function: $\sum \limits_{k=1}^n \left\lfloor \frac{n}{k} \right\rfloor \varphi(k) = \frac{n(n+1)}{2}$

Let $\varphi(n)$ be Euler's totient function, the number of positive integers less than or equal to $n$ and relatively prime to $n$. Challenge: Prove $$\sum_{k=1}^n \left\lfloor \frac{n}{k} ...
5
votes
1answer
223 views

Sum involving the hypergeometric function, power and factorial functions

I am finding some trouble in calculating the following sum involving the hypergeometric function, power and factorial functions. $$ \sum_{y=1}^\infty \mathrm{e}^z \cdot {}_1F_1\left(1-y;2;-z\right) ...
5
votes
3answers
247 views

Closed form for the sum of even fibonacci numbers?

I recently took a look a project euler, and I am trying to think of a smart way to do number 2. I looked at the sequence, and I saw that the question is basically asking for $$ \sum_{i=1}^n F_{3i} $$ ...
5
votes
1answer
1k views

Show $1 + 2 \sum_{n=1}^N \cos n x = \frac{ \sin (N + 1/2) x }{\sin \frac{x}{2}}$ for $x \neq 0$ [duplicate]

For $x \neq 0$, $$ 1 + 2 \sum_{n=1}^N \cos n x = \frac{ \sin (N + 1/2) x }{\sin \frac{x}{2}} $$
3
votes
2answers
124 views

Summation involving totient function: $\sum_{d\mid n} \varphi(d)=n$ [duplicate]

Prove that:$$\sum_{d\mid n} \varphi(d)=n$$ Where $\varphi(n)$ denotes the number of positive integers $m$ less than or equal to $n$ such that $\gcd(m,n)=1$ I am lost here, any help would be ...
2
votes
1answer
84 views

evaluating sum of limits [duplicate]

I was given this problem in definite integral and limits chapter $$\lim_{n \to \infty}\left( \frac{n}{n^2 + 1^2}+ \frac{n}{n^2 + 2^2} + \frac{n}{n^2 + 3^2 }+........+\frac{n}{n^2 + n^2}\right)$$ which ...
1
vote
3answers
345 views

Show that $f_0 - f_1 + f_2 - \cdots - f_{2n-1} + f_{2n} = f_{2n-1} - 1$ when $n$ is a positive integer

Letting $f_n$ be the Fibonacci numbers, show that $f_0 - f_1 + f_2 - \cdots - f_{2n-1} + f_{2n} = f_{2n-1} - 1$ when $n$ is a positive integer. Just some homework help. Need to prove. Thank you in ...
5
votes
5answers
1k views

Proving $\sum_{k=0}^{n}k{n\choose k}^2 = n{2n-1 \choose n-1} $

[Corrected question] I'm struggling at proving the following combinatorical identity: $$\sum_{k=0}^{n}k{n\choose k}^2 = n{2n-1 \choose n-1} $$ I would like to see a combinatorical (logical) solution, ...
5
votes
2answers
108 views

Sum up to number $N$ using $1,2$ and $3$

So the question asked was finding out the number of ways(combinations), a given number $N$ can be formed using the sum of $1,2$ or $3$. (eg) For $n = 8$, the answer is $10$ The given solution for ...
4
votes
2answers
86 views

Prove: $\sum_{k<n, (k,n)=1} k= \frac{1}{2}n \varphi (n)$

Prove: $\sum_{k<n, (k,n)=1}k = \frac{1}{2}n \varphi (n)$ I have had strep throat and missed the lecture discussing properties of the Euler function. Any help in solving this is appreciated. ...
3
votes
5answers
175 views

$\sum_{k=1}^nH_k = (n+1)H_n-n$. Why?

This is motivated by my answer to this question. The Wikipedia entry on harmonic numbers gives the following identity: $$ \sum_{k=1}^nH_k=(n+1)H_n-n $$ Why is this? Note that I don't just ...
2
votes
2answers
149 views

Alternating sum of binomial coefficients $\sum(-1)^k{n\choose k}\frac{1}{k+1}$ [duplicate]

I would appreciate if somebody could help me with the following problem Q:Calculate the sum: $$ \sum_{k=1}^n (-1)^k {n\choose k}\frac{1}{k+1} $$
0
votes
3answers
81 views

Derive Closed form sum of N^2

Can anyone explain to me how you would derive this ? I have this question asked in a CS class and can't figure out how to derive it. it has to be derived as you would with sum of N ex ...
0
votes
2answers
133 views

How to simplify this summation containing floor

Here's the summation: $$ S = \sum_{i=1}^{n}\left\lfloor\frac{n}{i}\right\rfloor $$ It can also be written as $$ S = \sum_{i=1}^{n}\left\lfloor\frac{n- (n \mod i)}{i}\right\rfloor $$ The answer will ...
8
votes
1answer
376 views

Apéry's constant ($\zeta(3)$) value

I tried to find some proofs about the Apéry's constant, but I didn't find any intuitive proof. Is this constant given by the "brutal force" summing of $1 + \frac{1}{2^3} + \frac{1}{3^3} + ...
4
votes
2answers
58 views

Finding the summation of the floor of the series identity

I would appreciate if somebody could help me with the following problem: Q: How to proof ? The number of positive divisors of $n$ is denoted by $d(n)$ ...
3
votes
3answers
339 views

Sum of all real number for any interval.

We know that sum of natural numbers over any interval always exists. For example sum from 0 to 10 of all natural numbers is $$S=\sum_{n=0}^{10}{n}=\frac{0+10}{2}\times{10}=55$$ But what about real ...
3
votes
2answers
188 views

Sum of derangements and binomial coefficients

I'm trying to find the closed form for the following formula $$\sum_{i=0}^n {n \choose i} D(i)$$ where $D(i)$ is the number of derangement for $i$ elements. A derangement is a permutation in which ...
3
votes
4answers
249 views

How $\frac{1}{n}\sum_{i=1}^n X_i^2 - \bar X^2 = \frac{\sum_{i=1}^n (X_i - \bar X)^2}{n}$

How $\frac{1}{n}\sum_{i=1}^n X_i^2 - \bar X^2 = \frac{\sum_{i=1}^n (X_i - \bar X)^2}{n}$ i have tried to do that by the following procedure: $\frac{1}{n}\sum_{i=1}^n X_i^2 - \bar X^2$ ...
3
votes
2answers
2k views

Compute $1^2 + 3^2+ 5^2 + \cdots + (2n-1)^2$ by mathematical induction

I am doing mathematical induction. I am stuck with the question below. The left hand side is not getting equal to the right hand side. Please guide me how to do it further. $1^2 + 3^2+ 5^2 + ...
2
votes
1answer
42 views

Compute summation with a relative error of O(n^-2)

$a(n) = \sum_{i \geq 0} a_i n^{-i}$, how can we compute the value of $a(n)^n$ with a relative error of $O(n^{-2})$?
2
votes
0answers
62 views

Find Gcd summation fast?

Find the value of the summation: $$ val=\left( \sum_{i=1}^a \sum_{j=1}^b\sum_{k=1}^c....\sum_{x=1}^p GCD(i,j,k,..x) \right)$$ Contraints $2\leq$number of summation terms$\leq 500$, $1\leq ...
2
votes
3answers
505 views

For the Fibonacci numbers, show for all $n$: $F_1^2+F_2^2+\dots+F_n^2=F_nF_{n+1}$

The definition of a Fibonacci number is as follows: $$F_0=0\\F_1=1\\F_n=F_{n-1}+F_{n-2}\text{ for } n\geq 2$$ Prove the given property of the Fibonacci numbers for all n greater than or equal to 1. ...
2
votes
3answers
100 views

How would you prove $\sum_{i=1}^{n} (3/4^i) < 1$ by induction?

How would you prove this by induction? $\sum_{i=1}^{n} (3/4^i) < 1$, $ \forall n \geq 2$ I can do the base case but dont know how to to finish it
2
votes
3answers
285 views

How to solve second degree recurrence relation?

For first degree recurence relation it is as simple as $f(n)=a^n\cdot f(0)+b\dfrac{a^n-1}{a-1}$. But how do you solve second degree? For example $$f(n)=\begin{cases} 1,&\text{for }n=1\\ ...
1
vote
2answers
183 views

Is $1 + 2 + 3 + \dots = -\frac{1}{12}$ really true? [duplicate]

I've read this strange result of the sum of all positive integers being $-\frac{1}{12}$. Is it really true? Does this also mean this is true? $$\sum_{n=1}^k n = \frac{k\cdot(k+1)}{2}$$ ...
1
vote
1answer
98 views

Sum of self power

Is there a formula to calculate the sum of a number to the power of this same number, like: $$1^1 + 2^2 + 3^3 + 4^4 + 5^5 + ... + n^n$$? or $$x^x + (x+1)^{(x+1)} + (x+2)^{(x+2)} + ... + ...
1
vote
3answers
103 views

Proving $\sum_{k=1}^nk^3 = \left(\sum_{k=1}^n k\right)^2$ using complete induction [duplicate]

I tried to prove the following statement using complete induction but I couldn't manage to solve it because I got a complex notation eventually. The statement is the following: $$\sum_{k=1}^nk^3 = ...
1
vote
3answers
114 views

Induction proof of $\sum_{k=1}^{n} \binom n k = 2^n -1 $

Prove by induction: $$\sum_{k=1}^{n} \binom n k = 2^n -1 $$ for all $n\in \mathbb{N}$. Today I wrote calculus exam, I had this problem given. I have the feeling that I will get $0$ points for my ...
1
vote
4answers
120 views

Computing $\sum_{i=1}^{n}i^{k}(n+1-i)$

I dont know how to proceed with solving $$\sum_{i=1}^{n}i^{k}(n+1-i).$$ Please give advise.
0
votes
1answer
112 views

Meaning of $\sum_{4}^{0}$

Encountered this question in Knuth et al's concrete mathematics. The question is what does $\sum_4^0$ mean? I think it does not mean anything unless we assign it meaning. It's just notation that can ...
0
votes
1answer
52 views

Inequality with monotone functions on power set

Consider a discrete probability space $\left( S, F, P\right)$, where $S = \{ 1, 2, \ldots, N \}$. Consider the set $$S' := \mathcal{P}(S) \setminus \{ \varnothing\} = \{ \{ 1\}, \{ 2\}, \ldots, ...
0
votes
1answer
92 views

Summing a series with a changing power

$$\sum_{r=0}^{k-1}4^r$$ Hi, I was wondering whether anyone could explain how to work this out. I know the end result is $\frac{4^k-1}{3}$, but I don't know why or how to get there. Thank you :D
0
votes
1answer
167 views

Expected value of a Poisson sum of confluent hypergeometric functions

How to compute the expected value of a Poisson sum of the following confluent hypergeometric function: $$ \sum_{y=1}^{Y} {}_1F_1(y,1,z) $$ where y is positive integer taking values from the Poisson ...
0
votes
1answer
57 views

$\sum_{i=1}^{n} (3i + 2n)$

I want to verify what would be the simplified solved version of this summation. $$\sum_{i=1}^{n} (3i + 2n)$$ Would it be this? $$ \frac32n^2 + \frac32n + 2n^2 $$
16
votes
1answer
195 views

Closed form for $\sum_{n=1}^\infty\frac{(-1)^n n^a H_n}{2^n}$

Is there a closed form for the sum $$\sum_{n=1}^\infty\frac{(-1)^n n^a H_n}{2^n},$$ where $H_n$ are harmonic numbers: $$H_n=\sum_{k=1}^n\frac{1}{k}=\frac{\Gamma'(n+1)}{n!}+\gamma.$$ This is a ...
23
votes
1answer
509 views

Proving $\sum_{n=-\infty}^\infty e^{-\pi n^2} = \frac{\sqrt[4] \pi}{\Gamma\left(\frac 3 4\right)}$

Wikipedia informs me that $$S = \vartheta(0;i)=\sum_{n=-\infty}^\infty e^{-\pi n^2} = \frac{\sqrt[4] \pi}{\Gamma\left(\frac 3 4\right)}$$ I tried considering $f(x,n) = e^{-x n^2}$ so that its ...
14
votes
2answers
280 views

How prove this sum $\sum_{n=1}^{\infty}\binom{2n}{n}\frac{(-1)^{n-1}H_{n+1}}{4^n(n+1)}$

show that $$\sum_{n=1}^{\infty}\binom{2n}{n}\dfrac{(-1)^{n-1}H_{n+1}}{4^n(n+1)}=5+4\sqrt{2}\left(\log{\dfrac{2\sqrt{2}}{1+\sqrt{2}}}-1\right)$$ where ...
17
votes
2answers
306 views

How to either prove or disprove if it is possible to arrange a series of numbers such the sum of any two adjacent number adds up to a prime number

I'm wondering if it's possible to write a theorem to prove or disprove the possibility of arranging a sequence of numbers (1,2,...n) such that the sum of any two numbers adds up to a prime number. An ...
13
votes
1answer
475 views

How to evaluate $\int_{0}^{1}{\frac{{{\ln }^{2}}\left( 1-x \right){{\ln }^{2}}\left( 1+x \right)}{1+x}dx}$

I want to evaluate $$\int_{0}^{1}{\frac{{{\ln }^{2}}\left( 1-x \right){{\ln }^{2}}\left( 1+x \right)}{1+x}dx}$$ I run this integral on Maple, It does converge. How we get a closed form? Is that ...
17
votes
4answers
397 views

How find this sum $I_n=\sum_{k=0}^{n}\frac{H_{k+1}H_{n-k+1}}{k+2}$

$$I_n=\sum_{k=0}^{n}\dfrac{H_{k+1}H_{n-k+1}}{k+2}$$ where $$H_{n}=1+\dfrac{1}{2}+\cdots+\dfrac{1}{n}$$ my try:since ...
17
votes
1answer
727 views

Prove that sum is finite

Let $j \in \mathbb{N}$. Set $$ a_j^{(1)}=a_j:=\sum_{i=0}^j\frac{(-1)^{j-i}}{i!6^i(2(j-i)+1)!} $$ and $a_j^{(l+1)}=\sum_{i=0}^ja_ia_{j-i}^{(l)}$. Please help me to prove that the following sum is ...