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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.

128
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5answers
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How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression?

How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression? For example here is the sum of $\cos$ series: $$\sum_{k=0}^{n-1}\cos (a+k \cdot d) =\frac{\sin(n \times \frac{...
57
votes
12answers
32k views

Proving $1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$ using induction

How can I prove that $$1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$$ for all $n \in \mathbb{N}$? I am looking for a proof using mathematical induction. Thanks
111
votes
5answers
28k views

Value of $\sum\limits_n x^n$

Why does the following hold: \begin{equation*} \displaystyle \sum\limits_{n=0}^{\infty} 0.7^n=\frac{1}{1-0.7} = 10/3 ? \end{equation*} Can we generalize the above to $\displaystyle \sum_{n=0}^{\...
83
votes
30answers
164k views

Proof for formula for sum of sequence $1+2+3+\ldots+n$?

Apparently $1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2$. How? What's the proof? Or maybe it is self apparent just looking at the above? PS: This problem is know as "The sum of the first $n$ positive ...
44
votes
13answers
8k views

Proof of the Hockey-Stick Identity: $\sum\limits_{t=0}^n \binom tk = \binom{n+1}{k+1}$

After reading this question, the most popular answer use the identity $$\sum_{t=0}^n \binom{t}{k} = \binom{n+1}{k+1}.$$ What's the name of this identity? Is it the identity of the Pascal's triangle ...
11
votes
1answer
6k views

$\sum \cos$ when angles are in arithmetic progression [duplicate]

Possible Duplicate: How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression? Prove $$\cos(\alpha) + \cos(\alpha + \beta) + \cos(\alpha + 2\beta) + \dots + \cos[\...
113
votes
31answers
57k views

Sum of First $n$ Squares Equals $\frac{n(n+1)(2n+1)}{6}$

I am just starting into calculus and I have a question about the following statement I encountered while learning about definite integrals: $$\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}$$ I really ...
149
votes
23answers
12k views

Proving the identity $\sum_{k=1}^n {k^3} = \big(\sum_{k=1}^n k\big)^2$ without induction

I recently proved that $$\sum_{k=1}^n k^3 = \left(\sum_{k=1}^n k \right)^2$$ using mathematical induction. I'm interested if there's an intuitive explanation, or even a combinatorial interpretation ...
60
votes
6answers
45k views

Combinatorial proof of summation of $\sum\limits_{k = 0}^n {n \choose k}^2= {2n \choose n}$

I was hoping to find a more "mathematical" proof, instead of proving logically $\displaystyle \sum_{k = 0}^n {n \choose k}^2= {2n \choose n}$. I already know the logical Proof: $${n \choose k}^2 = {...
11
votes
4answers
2k views

Prove $\sum\limits_{i=0}^n\binom{i+k-1}{k-1}=\binom{n+k}{k}$ (a.k.a. Hockey-Stick Identity) [duplicate]

Let $n$ be a nonnegative integer, and $k$ a positive integer. Could someone explain to me why the identity $$ \sum_{i=0}^n\binom{i+k-1}{k-1}=\binom{n+k}{k} $$ holds?
7
votes
9answers
2k views

How to compute the formula $\sum \limits_{r=1}^d r \cdot 2^r$?

Given $$1\cdot 2^1 + 2\cdot 2^2 + 3\cdot 2^3 + 4\cdot 2^4 + \cdots + d \cdot 2^d = \sum_{r=1}^d r \cdot 2^r,$$ how can we infer to the following solution? $$2 (d-1) \cdot 2^d + 2. $$ Thank you
17
votes
10answers
8k views

How to prove this binomial identity $\sum_{r=0}^n {r {n \choose r}} = n2^{n-1}$?

I am trying to prove this binomial identity $\sum_{r=0}^n {r {n \choose r}} = n2^{n-1}$ but am not able to think something except induction,which is of-course not necessary (I think) here, so I am ...
93
votes
5answers
22k views

The sum of an uncountable number of positive numbers

Claim:If $(x_\alpha)_{\alpha\in A}$ is a collection of real numbers $x_\alpha\in [0,\infty]$ such that $\sum_{\alpha\in A}x_\alpha<\infty$, then $x_\alpha=0$ for all but at most countably many $\...
66
votes
4answers
6k views

Identity for convolution of central binomial coefficients: $\sum\limits_{k=0}^n \binom{2k}{k}\binom{2(n-k)}{n-k}=2^{2n}$

It's not difficult to show that $$(1-z^2)^{-1/2}=\sum_{n=0}^\infty \binom{2n}{n}2^{-2n}z^{2n}$$ On the other hand, we have $(1-z^2)^{-1}=\sum z^{2n}$. Squaring the first power series and comparing ...
9
votes
7answers
5k views

Alternating sum of binomial coefficients: given $n \in \mathbb N$, prove $\sum^n_{k=0}(-1)^k {n \choose k} = 0$

Let $n$ be a positive integer. Prove that \begin{align} \sum_{k=0}^n \left(-1\right)^k \binom{n}{k} = 0 . \end{align} I tried to solve it using induction, but that got me nowhere. I think the ...
31
votes
12answers
22k views

Why $\sum_{k=1}^{\infty} \frac{k}{2^k} = 2$? [duplicate]

Can you please explain why $$ \sum_{k=1}^{\infty} \dfrac{k}{2^k} = \dfrac{1}{2} +\dfrac{ 2}{4} + \dfrac{3}{8}+ \dfrac{4}{16} +\dfrac{5}{32} + \dots = 2 $$ I know $1 + 2 + 3 + ... + n = \dfrac{n(n+1)}{...
20
votes
5answers
5k views

Algebraic Proof that $\sum\limits_{i=0}^n \binom{n}{i}=2^n$

I'm well aware of the combinatorial variant of the proof, i.e. noting that each formula is a different representation for the number of subsets of a set of $n$ elements. I'm curious if there's a ...
34
votes
5answers
10k views

How to find the sum of this series : $1+\frac{1}{2}+ \frac{1}{3}+\frac{1}{4}+\dots+\frac{1}{n}$

Problem : How to find the sum of this series : $1+\frac{1}{2}+ \frac{1}{3}+\frac{1}{4}+\dots+\frac{1}{n}$ This is a Harmonic progression : So is this formula correct to sum the series : $\frac{(...
32
votes
4answers
2k views

A closed form of $\sum_{k=0}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right)$

I am looking for a closed form of the following series \begin{equation} \mathcal{I}=\sum_{k=1}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right) \end{equation} I have no idea how to ...
8
votes
4answers
574 views

Proving Binomial Identity without calculus

How to establish the following identities without the help of calculus: For positive integer $n, $ $$\sum_{1\le r\le n}\frac{(-1)^{r-1}\binom nr}r=\sum_{1\le r\le n}\frac1r $$ and $$\sum_{0\le r\le ...
6
votes
3answers
896 views

Evaluating Combination Sum $\sum{n+k\choose 2k} 2^{n-k}$

Evaluate $$\sum_{k=0}^n{n+k\choose 2k} 2^{n-k}$$ So im not really sure how to begin with this. I would imagine we start with dividing out $2^{n}$, but not really sure much past that
13
votes
6answers
6k views

Proving $\sum\limits_{k=1}^{n}{\frac{1}{\sqrt{k}}\ge\sqrt{n}}$ with induction

I am just starting out learning mathematical induction and I got this homework question to prove with induction but I am not managing. $$\sum\limits_{k=1}^{n}{\frac{1}{\sqrt{k}}\ge\sqrt{n}}$$ ...
38
votes
8answers
4k views

why is $\sum\limits_{k=1}^{n} k^m$ a polynomial with degree $m+1$ in $n$

why is $\sum\limits_{k=1}^{n} k^m$ a polynomial with degree $m+1$ in $n$? I know this is well-known. But how to prove it rigorously? Even mathematical induction does not seem so straight-forward. ...
11
votes
6answers
2k views

Expressing a factorial as difference of powers: $\sum_{r=0}^{n}\binom{n}{r}(-1)^r(l-r)^n=n!$?

The successive difference of powers of integers leads to factorial of that power. Here's the formula: $$\sum_{r=0}^{n}\binom{n}{r}(-1)^r(n-r)^n=n!$$ Can anyone give a proof of this result? Note: ...
20
votes
7answers
6k views

Floor function properties: $[2x] = [x] + [ x + \frac12 ]$ and $[nx] = \sum_{k = 0}^{n - 1} [ x + \frac{k}{n} ] $

I'm doing some exercises on Apostol's calculus, on the floor function. Now, he doesn't give an explicit definition of $[x]$, so I'm going with this one: DEFINITION Given $x\in \Bbb R$, the integer ...
7
votes
4answers
1k views

What is the formula for $\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+\cdots +\frac{1}{n(n+1)}$

How can I find the formula for the following equation? $$\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+\cdots +\frac{1}{n(n+1)}$$ More importantly, how would you approach finding the ...
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 ...
6
votes
6answers
9k views

How to prove Vandermonde's Identity: $\sum_{k=0}^{n}\binom{R}{k}\binom{M}{n-k}=\binom{R+M}{n}$?

How can we prove that $$\sum_{k=0}^{n}\binom{R}{k}\binom{M}{n-k}=\binom{R+M}{n}?$$ (Presumptive) Source: Theoretical Exercise 8, Ch 1, A First Course in Probability, 8th ed by Sheldon Ross.
57
votes
4answers
44k views

What is the term for a factorial type operation, but with summation instead of products? [duplicate]

(Pardon if this seems a bit beginner, this is my first post in math - trying to improve my knowledge while tackling Project Euler problems) I'm aware of Sigma notation, but is there a function/name ...
25
votes
2answers
127k 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 ...
11
votes
4answers
9k views

Proving $\sum\limits_{k=0}^{n}\cos(kx)=\frac{1}{2}+\frac{\sin(\frac{2n+1}{2}x)}{2\sin(x/2)}$

I am being asked to prove that $$\sum\limits_{k=0}^{n}\cos(kx)=\frac{1}{2}+\frac{\sin(\frac{2n+1}{2}x)}{2\sin(x/2)}$$ I have some progress made, but I am stuck and could use some help. What I did: ...
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, ...
9
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5answers
10k 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 hints ...
23
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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, ...
3
votes
2answers
3k views

Proving $\sum\limits_{i=0}^n i 2^{i-1} = (n+1) 2^n - 1$ by induction

I'm trying to apply an induction proof to show that $((n+1) 2^n - 1 $ is the sum of $(i 2^{i-1})$ from $0$ to $n$. the base case: L.H.S = R.H.S we assume that $(k+1) 2^k - 1 $ is true. we need to ...
2
votes
2answers
2k views

Proving the geometric sum formula by induction

$$\sum_{k=0}^nq^k = \frac{1-q^{n+1}}{1-q}$$ I want to prove this by induction. Here's what I have. $$\frac{1-q^{n+1}}{1-q} + q^{n+1} = \frac{1-q^{n+1}+q^{n+1}(1-q)}{1-q}$$ I wanted to factor a $q^{...
27
votes
13answers
3k 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 ...
14
votes
2answers
4k views

Finite Sum of Power?

Can someone tell me how to get a closed form for $$\sum_{k=1}^n k^x$$ For $x = 1$, it's just the classic $\frac{n(n+1)}2$. What is it for $x > 1$?
24
votes
7answers
2k 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 ...
12
votes
8answers
2k views

I found this odd relationship, $x^2 = \sum_\limits{k = 0}^{x-1} (2k + 1)$. [duplicate]

I stumbled across this relationship while I was messing around. What's the proof, and how do I understand it intuitively? It doesn't really make sense to me that the sum of odd numbers up to $2x + 1$ ...
19
votes
8answers
13k views

Proving $\sum_{k=0}^n(-1)^k\binom nk=0$

Show that $$\sum_{k=0}^n(-1)^k\binom nk=0$$ So for odd $n$ we have an even number of terms. So $\binom nk=\binom n{n-k}$ which have opposite signs. Thus the sum is 0. For even $n$ we have that $$\...
12
votes
2answers
8k views

Inductive proof that ${2n\choose n}=\sum{n\choose i}^2.$

I would like to prove inductively that $${2n\choose n}=\sum_{i=0}^n{n\choose i}^2.$$ I know a couple of non-inductive proofs, but I can't do it this way. The inductive step eludes me. I tried naively ...
11
votes
2answers
53k views

Proving by induction that $ \sum_{k=0}^n{n \choose k} = 2^n$

Prove by induction that for all $n \ge 0$: $${n \choose 0} + {n \choose 1} + ... + {n \choose n} = 2^n.$$ In the inductive step, use Pascal’s identity, which is: $${n+1 \choose k} = {n \choose k-1} ...
54
votes
3answers
53k views

$\sum k! = 1! +2! +3! + \cdots + n!$ ,is there a generic formula for this?

I came across a question where I needed to find the sum of the factorials of the first $n$ numbers. So I was wondering if there is any generic formula for this? Like there is a generic formula for ...
28
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 ...
1
vote
7answers
276 views

Proof that $\left(\sum^n_{k=1}x_k\right)\left(\sum^n_{k=1}y_k\right)\geq n^2$

If $x_1,...,x_n$ are positive real numbers and if $y_k=1/x_k$, prove that $$\left(\sum^n_{k=1}x_k\right)\left(\sum^n_{k=1}y_k\right)\geq n^2.$$ I've been learning induction, and I've come across this ...
4
votes
5answers
2k views

Sum with binomial coefficients: $\sum_{k=0}^{n}{2n\choose 2k}$

I'm repeating material for test and I came across the example that I can not do. How to calculate this sum: $\displaystyle\sum_{k=0}^{n}{2n\choose 2k}$?
4
votes
2answers
342 views

Find $\sum_{k=1}^{\infty}\frac{1}{2^{k+1}-1}$

Calculate $$\sum_{k=1}^{\infty}\frac{1}{2^{k+1}-1}.$$ I used Wolfram|Alpha to compute it and got it to be approximately equal to $0.6$. How to compute it? Can someone give me a hint or a suggestion ...
302
votes
8answers
30k views

The length of toilet roll

Fun with Math time. My mom gave me a roll of toilet paper to put it in the bathroom, and looking at it I immediately wondered about this: is it possible, through very simple math, to calculate (with ...
37
votes
6answers
2k views

Finite Sum $\sum\limits_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}$

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 $(\star)$ by ...