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Questions tagged [summation]

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

31
votes
4answers
1k views

How close can $\sum_{k=1}^n \sqrt{k}$ be to an integer?

How close can $S(n) = \sum_{k=1}^n \sqrt{k}$ be to an integer? Is there some $f(n)$ such that, if $I(x)$ is the closest integer to $x$, then $|S(n)-I(S(n))|\ge f(n)$ (such as $1/n^2$, $e^{-n}$, ...). ...
30
votes
5answers
9k views

What is $\sum\limits_{i=1}^n \sqrt i\ $?

What is $\sum\limits_{i=1}^n\sqrt i\ $? Also I noticed that $\sum\limits_{i=1}^ni^k=P(n)$ where $k$ is a natural number and $P$ is a polynomial of degree $k+1$. Does that also hold for any real ...
30
votes
2answers
5k views

How do you calculate a sum over a polynomial?

I know that given a polynomial $p(i)$ of degree $d$, the sum $\sum_{i=0}^n p(i)$ would have a degree of $d + 1$. So for example $$ \sum_{i=0}^n \left(2i^2 + 4\right) = \frac{2}{3}n^3+n^2+\frac{13}{3}...
30
votes
3answers
1k views

Proof/derivation of $\lim\limits_{n\to\infty}{\frac1{2^n}\sum\limits_{k=0}^n\binom{n}{k}\frac{an+bk}{cn+dk}}\stackrel?=\frac{2a+b}{2c+d}$?

I just came up with the following identity while solving some combinatorial problem but not sure if it's correct. I've done some numerical computations and they coincide. $$\lim_{n\to \infty}{\frac{1}{...
29
votes
2answers
1k views

Trig sum: $\tan ^21^\circ+\tan ^22^\circ+…+\tan^2 89^\circ = ?$

As the title suggests, I'm trying to find the sum $$\tan^21^\circ+\tan^2 2^\circ+...+\tan^2 89^\circ$$ I'm looking for a solution that doesn't involve complex numbers, or any other advanced branch in ...
29
votes
5answers
6k views

use of $\sum $ for uncountable indexing set

I was wondering whether it makes sense to use the $\sum $ notation for uncountable indexing sets. For example it seems to me it would not make sense to say $$ \sum_{a \in A} a \quad \text{where A is ...
29
votes
1answer
875 views

Is there a closed form for this sum?

While generalizing the previous result, I conjectured that the series expansion of \begin{align*} \int_{0}^{\frac{\pi}{2}} \arctan \left( \frac{2x \sin\theta}{1-x^{2}} \right) \arctan \left( \frac{2y ...
29
votes
2answers
803 views

If $\left(1^a+2^a+\cdots+n^{a}\right)^b=1^c+2^c+\cdots+n^c$ for some $n$, then $(a,b,c)=(1,2,3)$?

Question : Is the following conjecture true? Conjecture : Let $a,b(\ge 2),c,n(\ge 2)$ be natural numbers. If $$\left(\sum_{k=1}^nk^a\right)^b=\sum_{k=1}^nk^c\ \ \ \ \ \cdots(\star)$$ for some $n$, ...
28
votes
1answer
2k views

Sum of Squares of Harmonic Numbers

Let $H_n$ be the $n^{th}$ harmonic number, $$ H_n = \sum_{i=1}^{n} \frac{1}{i} $$ Question: Calculate the following $$\sum_{j=1}^{n} H_j^2.$$ I have attempted a generating function approach but ...
28
votes
2answers
641 views

How to prove $\sum_{n=0}^{\infty} \frac{1}{1+n^2} = \frac{\pi+1}{2}+\frac{\pi}{e^{2\pi}-1}$

How can we prove the following $$\sum_{n=0}^{\infty} \dfrac{1}{1+n^2} = \dfrac{\pi+1}{2}+\dfrac{\pi}{e^{2\pi}-1}$$ I tried using partial fraction and the famous result $$\sum_{n=0}^{\infty} \...
27
votes
4answers
92k views

How many distinct functions can be defined from set A to B?

In my discrete mathematics class our notes say that between set $A$ (having $6$ elements) and set $B$ (having $8$ elements), there are $8^6$ distinct functions that can be formed, in other words: $|b|^...
27
votes
13answers
4k views

Methods to compute $\sum_{k=1}^nk^p$ without Faulhaber's formula

As far as every question I've seen concerning "what is $\sum_{k=1}^nk^p$" is always answered with "Faulhaber's formula" and that is just about the only answer. In an attempt to make more interesting ...
27
votes
4answers
1k views

Double summation with improper integral

So my friend sent me this really interesting problem. It goes: Evaluate the following expression: $$ \sum_{a=2}^\infty \sum_{b=1}^\infty \int_{0}^\infty \frac{x^{b}}{e^{ax} \ b!} \ dx .$$ Here is ...
26
votes
10answers
4k views

Explain non-convergent sums to a bright high schooler.

I know the higher-math answer to this question, but I'm asking how on Earth to explain it to a bright high school student. Here's the question, paraphrased: “I can see that $\left(1+x+x^{2}+x^{3}+x^{...
26
votes
3answers
859 views

How does one [easily] calculate $\sum\limits_{n=1}^\infty\frac{\mathrm{pop}(n)}{n(n+1)}$?

How does one [easily] calculate $\sum\limits_{n=1}^\infty\frac{\mathrm{pop}(n)}{n(n+1)}$, where $\mathrm{pop}(n)$ counts the number of bits '1' in the binary representation of $n$? Is there any trick ...
26
votes
3answers
769 views

Closed form for $\sum_{n=1}^\infty\frac{(-1)^n n^4 H_n}{2^n}$

Please help me to find a closed form for the sum $$\sum_{n=1}^\infty\frac{(-1)^n n^4 H_n}{2^n},$$ where $H_n$ are harmonic numbers: $$H_n=\sum_{k=1}^n\frac{1}{k}=\frac{\Gamma'(n+1)}{n!}+\gamma.$$
26
votes
5answers
892 views

Evaluating the sums $\sum\limits_{n=1}^\infty\frac{1}{n \binom{kn}{n}}$ with $k$ a positive integer

How to evaluate the sums $\sum\limits_{n=1}^\infty\frac{1}{n \binom{kn}{n}}$ with $k$ a positive integer? For $k=1$, the series does not converge. When $k=2$, I can prove that: $$\sum_{n=1}^\infty\...
26
votes
1answer
697 views

Finite sum $\sum_{n=2}^N\frac{1}{n^2}\sin^2(\pi x)\csc^2(\frac{\pi x}{n})$

I was looking for a closed form but it seemed too difficult. Now I'm seeking help to simplify this sum. The 50 bounty points or more will be awarded for any meaningful simplification of this sum. I ...
25
votes
3answers
745 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 $H_{n}=1+\dfrac{1}{2}+\dfrac{1}{3}+\cdots+\dfrac{...
25
votes
2answers
128k views

How to get to the formula for the sum of squares of first n numbers? [duplicate]

Possible Duplicate: How do I come up with a function to count a pyramid of apples? Proof that $\sum\limits_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$? Finite Sum of Power? I know that the sum of ...
25
votes
2answers
601 views

$\pi$ in terms of $4$?

I'm trying to define $\pi$ in terms of $4$ by placing a unit circle inside a square, and subtracting the corners of the square. I'm attempting to use summation to define the area of a corner, then ...
25
votes
2answers
639 views

$ \displaystyle \lim_{x \to 0+} \sum_{n=0}^\infty \frac{(-1)^n}{n!^x} = \frac{1}{2}$

Prove that $$ \lim_{x \to 0+} \sum_{n=0}^\infty \frac{(-1)^n}{n!^x} = \frac{1}{2}. $$ We know that $$ \sum_{n=0}^\infty \frac{(-1)^n}{n!^x}$$ converges for any $x>0$. So I try to evaluate the ...
24
votes
6answers
6k views

What is the sum of all positive even divisors of 1000?

I know similar questions and answers have been posted here, but I don't understand the answers. Can anyone show me how to solve this problem in a simple way? This is a math problem for 8th grade ...
24
votes
10answers
25k views

How to determine equation for $\sum_{k=1}^n k^3$

How do you find an algebraic formula for $\sum_{k=1}^n k^3$? I am able to find one for $\sum_{k=1}^n k^2$, but not $k^3$. Any hints would be appreciated.
24
votes
3answers
4k views

Does uncountable summation, with a finite sum, ever occur in mathematics?

Obviously, “most” of the terms must cancel out with opposite algebraic sign. You can contrive examples such as the sum of the members of R being 0, but does an uncountable sum, with a finite sum, ...
24
votes
7answers
3k views

Is there a direct, elementary proof of $n = \sum_{k|n} \phi(k)$?

If $k$ is a positive natural number then $\phi(k)$ denotes the number of natural numbers less than $k$ which are prime to $k$. I have seen proofs that $n = \sum_{k|n} \phi(k)$ which basically ...
24
votes
3answers
969 views

Finding the value of $\sum\limits_{k=0}^{\infty}\frac{2^{k}}{2^{2^{k}}+1}$

Does this weighted sum of reciprocals of Fermat numbers, $$ F=\sum_{k=0}^{\infty}\dfrac{2^{k}}{2^{2^{k}}+1} $$ have a nice closed form? Wolfram says it's $1$. Thanks.
24
votes
4answers
1k views

How to prove $\sum\limits_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$?

Other than the general inductive method,how could we show that $$\sum_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$$ Apart from induction, I tried with Wolfram Alpha to check the validity, ...
24
votes
2answers
2k views

A simple finite combinatorial sum I found, that seems to work, would have good reasons to work, but I can't find in the literature.

I was doing a consistency check for some calculations I'm performing for my master thesis (roughly - about a problem in discrete bayesian model selection) - and it turns out that my choice of priors ...
24
votes
5answers
650 views

Proving that $\gamma = \int_{0}^{1} \!\!\int_{0}^{1} \!\frac{x - 1}{(1 - x y) \log(x y)} \, \mathrm{d}{x} \, \mathrm{d}{y} $.

In 2005, J. Sondow found a surprising formula for the Euler-Mascheroni constant $ \gamma $. The formula is $$ \gamma = \int_{0}^{1} \int_{0}^{1} \frac{x - 1}{(1 - x y) \log(x y)} ~ \mathrm{d}{x} ~ ...
24
votes
4answers
73k views

How to prove a formula for the sum of powers of $2$ by induction?

How do I prove this by induction? Prove that for every natural number n, $ 2^0 + 2^1 + ... + 2^n = 2^{n+1}-1$ Here is my attempt. Base Case: let $ n = 0$ Then, $2^{0+1} - 1 = 1$ Which is true. ...
24
votes
1answer
860 views

Simplify $\left({\sum_{k=1}^{2499}\sqrt{10+{\sqrt{50+\sqrt{k}}}}}\right)\left({\sum_{k=1}^{2499}\sqrt{10-{\sqrt{50+\sqrt{k}}}}}\right)^{-1}$

Simplify $$\frac{\displaystyle\sum_{k=1}^{2499}\sqrt{10+{\sqrt{50+\sqrt{k}}}}}{\displaystyle\sum_{k=1}^{2499}\sqrt{10-{\sqrt{50+\sqrt{k}}}}}$$ I don't have any good idea. I need your help.
24
votes
1answer
711 views

Is this algebraic identity obvious? $\sum_{i=1}^n \prod_{j\neq i} {\lambda_j\over \lambda_j-\lambda_i}=1$

If $\lambda_1,\dots,\lambda_n$ are distinct positive real numbers, then $$\sum_{i=1}^n \prod_{j\neq i} {\lambda_j\over \lambda_j-\lambda_i}=1.$$ This identity follows from a probability calculation ...
24
votes
3answers
861 views

Mysterious identity

Playing around with Maple I found this identity $$\sum_{k=0}^{n-1}\frac{2k+1}{1-z^{2k+1}}=n\sum_{k=0}^{n-1}\frac{1}{1+z^{k}}$$ where $n$ is a positive integer, $z=\exp(\pi i/n)$. I was able to verify ...
24
votes
4answers
2k views

What are the applications of finite calculus

I'm reading through Concrete Mathematics [Graham, Knuth, Patashnik; 2nd edition], and in the section regarding Summation, they have a sub-section entitled "Finite and Infinite Calculus". In this ...
24
votes
1answer
652 views

Fibonacci-related sum

Related to this question Find a solution for $f\left(\frac{1}{x}\right)+f(x+1)=x$, what is this sum: $$\sum_{n=1}^{\infty}(-1)^n\left(\frac{F_n}{F_{n+1}}-\frac1{\phi}\right)$$ where $F_n$ is the $n$th ...
24
votes
1answer
1k views

Binomial Coefficients involving Prime Powers Minus 1

I would like to show the following is true; Let $p\in\mathbb{P}, \alpha,n\in\mathbb{N}$. Then $$p^\alpha\mid\sum_{k=1}^{n-1}\binom{(p^\alpha-1)n}{(p^\alpha -1)k}.$$ I've never worked with prime ...
23
votes
4answers
1k views

How find this $\sum_{n=1}^{\infty}\frac{H^3_{n}}{n+1}(-1)^{n+1}$

How Find this sum $$I=\sum_{n=1}^{\infty}\dfrac{H^3_{n}}{n+1}(-1)^{n+1}$$ where $H_{n}=1+\dfrac{1}{2}+\dfrac{1}{3}+\cdots+\dfrac{1}{n}$ My idea: since $$\dfrac{1}{n+1}(-1)^{n+1}=-\int_{-1}^{0}x^ndx$$...
23
votes
4answers
758 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 $$I_n=\dfrac{1+\dfrac{1}{2}+\cdots+\dfrac{1}{n+1}}{2}+\dfrac{\left(1+\dfrac{1}{2}\...
23
votes
2answers
612 views

New Year Maths 2018: $\color{green}{\binom ab+\binom bc+\binom cd}+\color{orange}{\binom de+\binom ef+\binom fg+\binom gh}=\color{red}{2018}$

Some maths to usher in the New Year! Find distinct integers $a,b,c,d,e,f,g,h$ such that $$\color{purple}{\binom ab+\binom bc}+\color{blue}{\binom cd}+\color{green}{\binom de+\binom ef}+\color{orange}{...
23
votes
2answers
425 views

How prove that:$\varphi(2)+\varphi(3)+\varphi(4)+\cdots+\varphi(n)\ge\frac{n(n-1)}{4}+1$

Prove that for $n\ge 3$, $$\varphi(2)+\varphi(3)+\varphi(4)+\cdots+\varphi(n)\ge\dfrac{n(n-1)}{4}+1$$ where $\varphi$ is the Euler's totient function I think we must use this $$\sum_{k=1}^{n}\varphi(...
22
votes
2answers
2k views

A Gift Problem for the Year 2018 [duplicate]

We had this problem in exam class yesterday on Combinatoric and it was supposed to be the new year gift from our teacher. The exercise was entitled A Gift Problem for the Year 2018 Problem: ...
22
votes
3answers
905 views

A (probably trivial) induction problem: $\sum_2^nk^{-2}\lt1$

So I'm a bit stuck on the following problem I'm attempting to solve. Essentially, I'm required to prove that $\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2} < 1$ for all $n$. I've been toiling ...
22
votes
5answers
2k views

Is there a combinatorial interpretation of the triangular numbers?

The triangular numbers count the number of items in a triangle with $n$ items on a side, like this: This can be calculated exactly by the formula $T_n = \sum_{k=1}^n k = \frac{n(n+1)}{2} = {n+1 \...
22
votes
5answers
675 views

Geometrical interpretation of $(\sum_{k=1}^n k)^2=\sum_{k=1}^n k^3$

Using induction it is straight forward to show $$\left(\sum_{k=1}^n k\right)^2=\sum_{k=1}^n k^3.$$ But is there also a geometrical interpretation that "proves" this fact? By just looking at those ...
22
votes
2answers
2k views

Integrating the formula for the sum of the first $n$ natural numbers

I was messing around with some math formulas today and came up with a result that I found pretty neat, and I would appreciate it if anyone could explain it to me. The formula for an infinite ...
22
votes
5answers
2k 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 ...
22
votes
5answers
11k views

A formula for the power sums: $1^n+2^n+\dotsc +k^n=\,$?

Is there explicit formula for the expression $1^n + 2^n + \dotsc + k^n\,$? I know that for $n=1$ the explicit formula becomes $S=k(k+1)/2$ and for $n=3$ the formula becomes $S^2$. But what about ...
22
votes
4answers
15k views

Summation Symbol: Changing the Order

I have some questions regarding the order of the summation signs (I have tried things out and also read the wikipedia page, nevertheless some questions remained unanswered): Original 1. wikipedia ...
22
votes
1answer
3k views

Combinatorial proof of a Fibonacci identity: $n F_1 + (n-1)F_2 + \cdots + F_n = F_{n+4} - n - 3.$

Does anyone know a combinatorial proof of the following identity, where $F_n$ is the $n$th Fibonacci number? $$n F_1 + (n-1)F_2 + \cdots + F_n = F_{n+4} - n - 3$$ It's not in the place I thought it ...