5
votes
3answers
121 views

Proof of Nesbitt's Inequality?

I just thought of this proof but I can't seem to get it to work. Let $a,b,c>0$, prove that $$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\ge \frac{3}{2}$$ Proof: Since the inequality is homogeneous, ...
1
vote
3answers
37 views

Summation Proof

I'm getting stuck halfway through this: Show that $$\sum_{i=1}^n (y_i - \bar y_s)^2 = \sum_{i=1}^n (y_i)^2 - n\bar y_s^2$$ My skills with manipulating sums are quite rusty. I multiply the left side ...
6
votes
5answers
127 views

Verify the following combinatorial identity: $\sum_{k=0}^{r} \binom{m}{k}\binom{n}{r-k} = \binom{m+n}{r}$

$$\sum_{k=0}^{r} \binom{m}{k}\binom{n}{r-k} = \binom{m+n}{r}$$ Nice, so I've proven some combinatorial identities before via induction, other more simple ones by committee selection models.... But ...
3
votes
0answers
51 views

Random Wolfram|Alpha identity $\sum_{k = 1}^{\infty}{\tan^{-1}}{\frac{1}{k^2}}$

I was watching a Numberphile video (on how $\tan^{-1}{1} + \tan^{-1}\frac{1}{2} + \tan^{-1}\frac{1}{3} = \frac{\pi}{2}$) and I thought about whether the series $$\sum_{k = ...
4
votes
1answer
46 views

Nested Sum Encountered in Maclaurin Expansion of $e^{-x^2}$

In the course of working out the Maclaurin expansions of $e^{-x^2}$ and $cos(x^2)$, I ran into the following nested sum: $$ \underbrace{ \sum_{a=0}^1 \left( a \sum_{b=0}^{a+1} b \left( ...
2
votes
4answers
64 views

Exotic proofs of $\sum_{j=0}^{n-1}\binom{p+j}{p}=\binom{p+n}{p+1}$

Let $p,n$ be positive integers. The following identity $\displaystyle \sum_{j=0}^{n-1}\binom{p+j}{p}=\binom{p+n}{p+1}$ may be proved by induction or by successive uses of Pascal's rule (both ...
5
votes
1answer
202 views

Proving these trigonometric sums $\sum\limits_{k=0}^{n-1}\sin\frac{2k^2\pi}{n}=\frac{\sqrt{n}}{2}\left(\cos\frac{n\pi}{2}-\sin\frac{n\pi}{2}+1\right)$

Can someone help me to prove that: $$ \sum_{k=0}^{n-1}\sin\frac{2k^2\pi}{n}=\frac{\sqrt{n}}{2}\left(\cos\frac{n\pi}{2}-\sin\frac{n\pi}{2}+1\right)$$ ...
1
vote
7answers
121 views

Error in proving of the formula the sum of squares

Given formula $$ \sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6} $$ And I tried to prove it in that way: $$ \sum_{k=1}^n (k^2)'=2\sum_{k=1}^n k=2(\frac{n(n+1)}{2})=n^2+n $$ $$ \int (n^2+n)\ \text d ...
1
vote
3answers
109 views

Prove: $\sum_{x=0}^{n} (-1)^x {n \choose x} = 0$

Is there a quick, fancy, way of proving sums such as this? Prove that: $$\sum_{x=0}^{n} (-1)^x {n \choose x} = 0$$ A recent homework assignment I turned in had a couple problems similar to the ...
2
votes
5answers
112 views

Proof via induction $1\cdot3 + 2\cdot4 + 3\cdot5 + \cdots + n(n+2) = \frac{n(n+1)(2n+7)}{6}$

(b) Prove that for every integer $n \ge 1$, $$1\cdot3 + 2\cdot4 + 3\cdot5 + \cdots + n(n+2) = \frac{n(n+1)(2n+7)}{6}$$ This is the second part of a two part question. Part (a) was the following: ...
0
votes
1answer
21 views

Proving claims about sequences by induction?

I am learning how to prove claims about finite sequences right now. Can you help me prove or disprove the following claim? ...
0
votes
0answers
13 views

Prove that $p_{k +1}$ = $P_0$$(\frac{b}{d})$$^{k+1}$ doesn't converge by using partial sum of geometric series

Prove that $p_{k +1}$ = $P_0$$(\frac{b}{d})$$^{k+1}$ does not converge by using the partial sum of the geometric series if the conditions are not met. I know that the condition is d > b where they ...
3
votes
6answers
776 views

How do I begin proving this binomial coefficient identity: ${n\choose 0} - {n\choose 1} + {n\choose 2} - {n\choose 3} + \dots = 0$

This is a homework question. I'm asked to prove the identity: $${n\choose 0} - {n\choose 1} + {n\choose 2} - {n\choose 3} + \dots = 0$$ (The sum ends with ${n\choose n} = 1$, with the sign of the ...
5
votes
3answers
973 views

Fibonacci trick and proving it. [duplicate]

I am trying to learn Fibonacci tricks and I have one that I can not prove. I know it works because Ive tried it multiple times but I have not a clue how to prove. Here it is: ...
0
votes
3answers
35 views

Induction summation proof

Don't want a full answer but can somebody help me in the right direction with this problem. Have to prove using induction $$\forall n \geqslant 2: \sum_{i=1}^{n} \frac{4}{5^{i}} < 1$$
7
votes
2answers
265 views

How to count matrices with rows and columns with an odd number of ones?

I proved that $\displaystyle \left(\sum_{k\, \rm odd}\binom{m}{k}\right)^{n-1}=\left(\sum_{k\;{\rm odd}}\binom{n}{k}\right)^{m-1}$ by counting matrices of size $n\times m$ with entries in $\{0,1\}$ ...
0
votes
3answers
106 views

Proofs Homework Help

I have been struggling with my proofs homework this week and would greatly appreciate any help. Prove, disprove, or give a counterexample: Suppose $f:X\to Y$, $A\subseteq Y$, $B\subseteq Y$ and ...
0
votes
3answers
96 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 ...
7
votes
3answers
438 views

How to find the sum of $i(i+1)\cdots(i+k)$ for fixed $k$ between $i = 1$ and $n$?

I learned that $$\sum \limits_{i=1}^n i(i+1) = \frac{n(n+1)(n+2)}{3}$$ or in general $$\sum \limits_{i = 1}^n i(i+1)(i+2) \dots (i + k) = \frac{n(n+1)\dots (n+k+1)}{k+2}$$ From a mathematical ...
11
votes
2answers
1k views

Why is this allowed? (“Fourier's Trick”; finding the coefficients in a Fourier Series)

In my textbook (Introduction to Electrodynamics, D. Griffiths), we derive the equation for some strange potential function. Eventually, we get to this (for $n \in \mathbb{Z}^+$): $$ V_0(y) = ...
1
vote
2answers
78 views

Proving a summation involving binomial coefficients.

I need to prove the following inductively: (http://upload.wikimedia.org/math/9/e/5/9e57871ba17c1ad48e01beb7e1bb3bb9.png) $$\sum_{i=1}^{n} i{n \choose i} = n2^{n-1}$$ And for the life of me I can't ...
1
vote
0answers
158 views

Choosing the vector that minimizes this sum related to the rearrangement inequality

The rearrangement inequality states that, for two sets of real numbers $x_1\leq\dots{}\leq x_n$ and $y_1\leq\dots{}\leq y_n$, the sum $\sum_{i=1}^n x_{\sigma(i)}y_i$ is minimized for the particular ...
3
votes
9answers
171 views
0
votes
1answer
31 views

In the process of proving Sum of Geometric Progression

I was reading the proof for the sum of geometric progression at http://www.proofwiki.org/wiki/Sum_of_Geometric_Progression and one of the statements is the following: ...
1
vote
1answer
192 views

Prove a summation inequality by induction

I was having trouble proving by induction with this problem. $$\sum_{i=1}^n \frac{3}{4^i} < 1$$ for all $n \geq 2$ I went to see my professor and he said try proving this equality ...
1
vote
1answer
43 views

How can I prove $ \sum_{i=0}^{k-1}x_{i}2^{i}\leq\sum_{i=0}^{k-1}2^{i}=2^{k}-1<2^{k} $

How can I prove this inequality property? Assumption: The values of $x_i$ can either be 0 or 1
5
votes
4answers
650 views

How to prove that $\sum_{k=0}^n \binom nk k^2=2^{n-2}(n^2+n)$ [duplicate]

I know that $$\sum_{k=0}^n \binom nk k^2=2^{n-2}(n^2+n),$$ but I cannot find a way how to prove it. I tried induction but it did not work. On wiki they say that I should use differentiation but I do ...
2
votes
3answers
104 views

How do you solve a recurrence with a summation function inside

Show that $$t(n) = 1 + \sum_{ j=0}^{n-1} t(j)$$ is the same as $$t(n) = 2^n$$ Initial condition $t(0) = 1$
1
vote
1answer
97 views

Where to find a proof of Silverman-Toeplitz?

I am referring to the theorem which gives a necessary and sufficient condition on a infinite matrix that maps convergent sequences to sequences converging to the same limits. Wiki gives a link to ...
1
vote
4answers
179 views

How would I go about showing $\cos(x+\frac{\pi}{2})=-\sin(x)$?

I want to look at the summation for both cos,sin.. $cos(x) = 1-\frac{x^2}{2!}+\frac{x^4}{4!}-...= \sum\nolimits_{i=0}^\infty \frac{(-1)^nx^{2n}}{(2n)!}$ so, ...
0
votes
2answers
476 views

Induction step for $\sum\limits_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6}$

I want to prove by induction that, $\sum\limits_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6}$ OK I got the initial step, however, I have problems with the induction step: Here is what I tried: ...
2
votes
0answers
62 views

Is my proof correct? (Also formally)

Hello dear community! I just worked on a problem in my discrete mathematics text book and wondered if my approach to a specific exercise is correct. There are no solutions to it, that's the reason I ...
4
votes
3answers
593 views

induction proof: $\sum_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$

I encountered the following induction proof on a practice exam for calculus: $$\sum_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$$ I have to prove this statement with induction. Can anyone please help me ...
3
votes
3answers
351 views

Evaluate $\displaystyle\sum_{k=1}^nk\cdot k!$

I discovered that the summation $\displaystyle\sum_{k=1}^n k\cdot k!$ equals $(n+1)!-1$. But I want a proof. Could anyone give me one please? Don't worry if it uses very advanced math, I can just ...
3
votes
0answers
212 views

Proving identity involving sum

I'm stuck trying to prove the following identity: $$\displaystyle\sum_{i=j}^k\frac{(-1)^{i+j}{k+1 \choose i}{i \choose j}(4S-i-k-3)!(4S-2i-1)}{(4S-i-j-1)!} $$ $$=\frac{(-1)^{j+k}(4S-2k-3)!{k+1 ...
3
votes
1answer
109 views

Proof the following trig series

Prove that $$\frac{ \sin x}{ \cos x}+\frac{\sin2x}{\cos^{2}x}+\frac{\sin3x}{\cos^{3}x}+\cdots+\frac{\sin nx}{\cos^{n}x}=\cot x-\frac{\cos(n+1)x}{\sin x \cos^{n}x}$$ I am not necessarily looking for a ...
0
votes
1answer
18 views

Clues to prove average in T is minor or equal than average in a smaller inner interval.

Suppose I want to prove (or disprove) this assertion Let $f$ be a discrete function, $T,h,k$ are constants So these terms are averages over $T$ and over $h$ $\sum\limits_{i=0}^{T}\frac {f(i)}{T}$ ...
1
vote
0answers
53 views

Visual/intuitive proof of why $\sum k^3 = (\sum k)^2$, where $k$ goes from 1 to $n$? [duplicate]

I understand that one could prove this by first proving the analytic expressions of the sigma terms through induction, and then square the $\displaystyle\sum_{k=1}^n k$ term to show LHS = RHS. Are ...
3
votes
2answers
842 views

Exponential function formula proof

How does one arrive at $e^4$ from $$\sum_{x=0}^{\infty}\frac{ 4^x}{x!}$$
2
votes
5answers
2k views

Proof via Induction for A Summation

I'm starting to understand how induction works (with the whole $k \to k+1$ thing), but I'm not exactly sure how summations play a role. I'm a bit confused by this question specifically: $$ ...
5
votes
3answers
206 views

Why is $\int\limits_{1}^{n} \log x \,dx \le \sum\limits_{x = 1}^{n}\log x$?

It has been a long time since I studied integrals, so this question may sound stupid. I was going through this wiki page, and came across the following inequality: $$\int_{1}^{n} \log x \,dx \le ...
1
vote
1answer
55 views

Simple Summation Proof with identities

Using some of the identities, determine the value of $\sum_0^5$ ${12 \choose i}$ . You may use the substitution ${12 \choose 6}$ = 924, but you may not evaluate the individual chooses. Proofs of ...
2
votes
2answers
58 views

proof by by directly manipulating the sum:

Prove that $$\sum_{i=0}^{n} x ^{i} = \frac{ 1-x ^{n+1} }{ 1-x }$$ to be used by directly manipulating the sum: let A be the sum, and show that xA = A + x^(n+1) -1 I don't get how its going to ...
0
votes
3answers
497 views

How to prove by induction that $\sum^n_{i=1}2^{i-1}=2^n-1$? [duplicate]

Possible Duplicate: How do I prove this by induction? (sum of powers of 2) Summation equation for $2^{x-1}$ How can I prove the following by induction? $$ \sum^n_{i=1}2^{i-1}=2^n-1 $$ I ...
37
votes
20answers
7k views

Proof that $\sum\limits_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$?

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 ...