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

learn more… | top users | synonyms

0
votes
0answers
27 views

Summation of series $\sum\limits_{i = 1}^n \sqrt{1 + 4i}$

How can I go about finding the following sum: $\sum\limits_{i = 1}^n \sqrt{1 + 4i}$
2
votes
0answers
23 views

Sums of binomial coefficients

Does anyone know something about the following sums? $$ S_m(n)=\sum\limits_{k=o}^n(-1)^k{mn\choose mk} $$ Notice that $S_m(n)=0$ for odd $n$, so we only consider $S_m(2n)$. It holds that $S_0(2n)=1$, ...
0
votes
1answer
21 views

In how many ways can you paint 90 distinct buckets?

In how many ways can you paint 90 distinct buckets, if 25 of them must be painted red, 40 of them must be painted blue, and 25 of them must be painted green? I am right to assume that these object ...
2
votes
2answers
33 views

The sum of binomial coefficients up to $k\le n/4$ does not exceed the $k$th coefficient

How would you prove the following (for when $k\leq\frac{n}{4}$)? $$\sum_{i=0}^{k-1} \binom ni \le \binom nk$$
1
vote
4answers
83 views

$\text{Prove that}$ $\frac{\sin(\frac{n+1}2)*\cos(\frac n2)}{\sin\frac 12} \ge\frac n2$

Prove that$$\frac{\sin\left(\frac{n+1}2\right)\times\cos\left(\frac n2\right)}{\sin\left(\frac 12\right)} \ge\frac n2$$ So far I've switched up the problem and gotten it down to all sin functions. I ...
-1
votes
2answers
37 views

What's the best way to find the sum of this sequence? [on hold]

I've got the following sequence: $$ 160-157+154-151+148-145+...+4-1 $$ How to find a sum of it?
3
votes
2answers
116 views

Sums $\sum_{n=1}^{N}\sqrt{4n+1}$

I need to find sum of the first N terms of the sequence whose nth term is as follow : T(n)= $\sqrt{4*n+1}$ So the sequence is : $\sqrt{5}$,$\sqrt{9}$,$\sqrt{13}$,$\sqrt{17}$,$\sqrt{21}$...... ...
0
votes
1answer
21 views

Proving an inequality on the sum of $\log$ of primes.

Let $S(x)=\sum_{p\leq x} \ln(p)$ where $\sum_{p\leq x}$ denotes a summation over the positive prime numbers that are $\leq x$ Prove that $\forall n \in \mathbb N, S(2n+1)-S(n+1)\leq ...
0
votes
0answers
29 views

Inequality in $\mathbb{Z}^2$

Let $k=(k_1,k_2)\in\mathbb{Z}^2$. Denote $|k|\leqslant n$ when $|k_1|,|k_2|\leqslant n$. I need help to show $$|\sum_{k+l+m=0}_{|k|,|l|,|m|\leqslant ...
1
vote
2answers
57 views

Calculate summation of square roots

Calculate summation of square roots i.e $$\sum_{i=1}^N\sqrt{i}$$ I tried to search for its formula on the net but I couldn't find any of its sources.
0
votes
0answers
7 views

Find $\sum_{i=1}^n \sum_{j=1}^m |a \cdot i-b\cdot j|$

Find $\sum_{i=1}^n \sum_{j=1}^m |a \cdot i-b\cdot j|$. There is a possibility that a closed form solution doesn't exist. In this case is there a non-trivial lower bound other than $0$? This question ...
0
votes
1answer
31 views

solve $\sum_{k=0}^a\binom{2a}{k}k$

solve $$\sum_{k=0}^a\binom{2a}{k}k$$ I solved $S=\sum_{k=0}^a\binom{2a}{k}$ using $\binom{2a}{k}=\binom{2a}{2a-k}$ and got $S=\frac{4^a+\binom{2a}{a}}{2}$ but this idea doesn't work with ...
1
vote
0answers
38 views

Is $\sum_{i=1}^n i = \sum_{i=n}^1 i$

When I enter these expressions into wolfram I get that they're not equal. Why is this? Essentially I'm trying to say $$ 1+2+\cdots+n = n+(n-1)+\cdots+1 $$
1
vote
1answer
40 views

Is there a tight upper bound on $\sum_{i=1}^n \sum_{j=1}^m \min(a\cdot i,b \cdot j)$

Is there a tight upper bound on $\sum_{i=1}^n \sum_{j=1}^m \min(a \cdot i,b \cdot j)$ for any $a,b \in \mathbb{R}^+$ For example one upper bound would be \begin{align} \sum_{i=1}^n \sum_{j=1}^m ...
0
votes
1answer
27 views

Giving a closed expression to $\sum_{i=0}^b (-1)^{b-i} \binom{b}{i}\frac{1}{a+b-i}$

I want to prove $\sum_{i=0}^b (-1)^{b-i} \binom{b}{i}\frac{1}{a+b-i} = \frac{(a-1)! b!}{(a+b)!}$ yet I feel like I don't know how to even approach this problem. Any hints are welcome.
5
votes
5answers
110 views

Calculate the binomial sum $ I_n=\sum_{i=0}^n (-1)^i { 2n+1-i \choose i} $

I need any hint with calculating of the sum $$ I_n=\sum_{i=0}^n (-1)^i { 2n+1-i \choose i}. $$ Maple give the strange unsimplified result $$ I_n={\frac {1/12\,i\sqrt {3} \left( - \left( \left( ...
1
vote
1answer
23 views

Sum of $n^{\text{th}}$ powers of roots of quadratic

How would I go about finding an expression (preferably closed form) for the sum of $\alpha^n+\beta^n$ in terms of $\alpha + \beta$ and $\alpha\beta$ (where $\alpha$ and $\beta$ are the roots of a ...
1
vote
0answers
37 views

Which numbers have the sum of their digits equal to the sum of the digits of their inverse?

$n$ is a number such as $n \in \mathbb{N}$ and $n >0$.(Eg. $n = 8$) $p$ is the sum of the digits of $n$ in base $10$.(Eg. $n=80$, $a = 8+0 = 8$) $q$ is the sum of the digits of $1/n$ in base ...
0
votes
0answers
18 views

Is it possible to derive the sum of the tangent or cotangent from this?

So we can suppose that we can find the finite sum $$\sum_{x=a}^b{ \tan{(x)} + \cot{(x)} } \tag{1}$$ for essentially all integer values of $x$. I'm wondering, can we derive either: $$\sum_{x=a}^b{ ...
1
vote
2answers
25 views

Can this recursive summation function be simplified?

I have the recursive function $$ a_n = \sum_{x=1}^n a_{n-x} a_{x-1} $$ where $a_0=1$ and $n$ is a positive integer. Looking at a graph of this function, it's very exponential in form, but it's not ...
0
votes
0answers
26 views

Sum of products of numbers in a list

For N numbers in a list, is there a formula to get the sum of products of the numbers in the list? Example: {3, 3, 2} = (3 * 3) + (3 * 2) + (3 * 2) Right now I'm ...
0
votes
1answer
23 views

Alternate expression for finite summation

"How many arithmetic operations are required to directly compute $$y=1+x+x^2+...+x^{1023}$$ Use a formula for the sum to come up with an alternate expression for $y$, and show that only 10 ...
1
vote
3answers
49 views

Calculate exact value of and infinite sum [duplicate]

Im trying to find the exact value of the infinite sum : 3 + 1/3 + 1/27 + 1/243 + 1/2187 + ... I can see that to generate new terms we take the previous term and divide by 9 or multiply by 9. Not ...
0
votes
2answers
53 views

Using induction to prove a formula for $\sin x+\sin 3x+\dots+\sin (2n-1)x$

I'm working from the text "Intro To Real Analysis" by William Trench. Here is what I have thus far. I will prove using Mathematical Induction that $\sin x+\sin 3x+...+\sin (2n-1)x=\frac{1-\cos ...
1
vote
2answers
34 views

Gosper's Identity $\sum_{k=0}^n{n+k\choose k}[x^{n+1}(1-x)^k+(1-x)^{n+1}x^k]=1 $

The page on Binomial Sums in Wolfram Mathworld http://mathworld.wolfram.com/BinomialSums.html (Equation 69) gives this neat-looking identity due to Gosper (1972): $$\sum_{k=0}^n{n+k\choose ...
1
vote
1answer
25 views

Proof that falling power can be converted to sum of normal powers

I'm trying to follow a proof that any falling power can be converted to a sum of multiples of regular powers, i.e. $x^{\underline{n} = \sum_{k=0}^n s_{n,k}x^k}$ with $s_{n,n}=1$ and $s_{n,0}=0$ for ...
8
votes
1answer
387 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 ...
1
vote
3answers
76 views

Proof of $\cos \theta+\cos 2\theta+\cos 3\theta+\cdots+\cos n\theta=\frac{\sin\frac12n\theta}{\sin\frac12\theta}\cos\frac12(n+1)\theta$

State the sum of the series $z+z^2+z^3+\cdots+z^n$, for $z\neq1$. By letting $z=\cos\theta+i\sin\theta$, show that $$\cos \theta+\cos 2\theta+\cos 3\theta+\cdots+\cos ...
0
votes
1answer
16 views

How do I evaluate $\sum _{ i=1 }^{ 2^n } (\frac{i}{2^n} - \frac{i-1}{2^n})(1-\frac{i-1}{2^n})$

What does this sum equal ? $\sum _{ i=1 }^{ 2^n } (\frac{i}{2^n} - \frac{i-1}{2^n})(1-\frac{i-1}{2^n})$ The answer I'm getting is $-\frac{1}{2^{n+1}} - \frac{1}{2^{2n+1}} + \frac{1}{2^{n}}$ but I ...
2
votes
2answers
55 views

Sum of a Series to Infinity

Evaluate the following sums: $\sum\limits_{i=0}^\infty\frac1{4^i}$. $\sum\limits_{i=0}^\infty\frac i{4^i}$. $\sum\limits_{i=0}^\infty\frac {i^2}{4^i}$. $\sum\limits_{i=0}^\infty\frac ...
-2
votes
1answer
39 views

Summation of Combination [closed]

PROVE $$\sum _{ t=0 }^{ r }{ { (-1) }^{ t } } {r \choose t}{ n-t \choose s}\quad =\quad { n-r \choose n-s }$$
0
votes
2answers
33 views

Evaluating sums question

I need to use the identity $\frac{1}{k^2-1}=\frac{1}{2}(\frac{1}{k-1}-\frac{1}{k+1})$ to evaluate $\sum_{k=2}^n \frac{1}{k^2-1}$ . I am confused about how to begin this proof. Also, how am I ...
0
votes
2answers
44 views

Einstein Summation - does the following equality hold: $a_{ij} x_i y_j = a_{ij} y_i x_j$

Does equality hold when $x_i = y_i$ and $x_j=y_j,$ and $ i, j = 1, ..., n $.
0
votes
3answers
45 views

(Taylor's theorem) Proving that $\sin(x) = \sum\limits_{n=0}^{\infty}\dfrac{(-1)^{n}x^{2n+1}}{(2n+1)!}$

I'm starting a class on Advanced Mathematics I next semester and I found a sheet of the class'es 2012 final exams, so I'm slowly trying to solve the exercises in it or find the general layout. I will ...
0
votes
2answers
31 views

Einstein Summation: How do I show $a_{ij} (x_i + y_j) \not= a_{ij}x_i + a_{ij}y_j $?

Einstein Summation: How do I show $a_{ij} (x_i + y_j) \not= a_{ij}x_i + a_{ij}y_j $?
0
votes
0answers
17 views

Transformation of a sum

I want to prove the following or a similar result: For $1\le k \le n$ \begin{align}&1-\sum\limits_{j=k+1}^n\binom nj(1-x)^jx^{n-j}~~~~~~(1)\\ ...
0
votes
1answer
41 views

what is $2\cdot 4\cdot 6\cdot 8 \cdot \ldots \cdot (2k+2)$?

I know that $2\cdot 4\cdot 6\cdot 8 \cdot \ldots \cdot (2k)$ is $2^kk!$ but what is the value of these terms up to the $(2k+2)^\text{th}$ term?
1
vote
3answers
38 views

Prove a sum formula by induction

I am to prove through induction that $$\sum_{k=1}^n (2k-1)^2 = \frac{n(2n-1)(2n+1)}{3}$$ And well, my method seems to be working, but I get stuck when I'm nearly done. First I prove the formula work ...
2
votes
4answers
59 views

Proving $\lim_{n \rightarrow \infty} \frac{\sum_{r=1}^{n} r^a}{n^{a+1}}=\frac{1}{a+1}$ [duplicate]

How do we prove that $$\lim_{n \rightarrow \infty} \dfrac{\displaystyle\sum_{r=1}^{n} r^a}{n^{a+1}}=\frac{1}{a+1}$$? This type of terms appear in problems on limits, but I am unable to prove this. ...
1
vote
2answers
35 views

Calculus add formula to derive new formula

I was asked to re-write a formula forward and backward and derive a new formula from it. Here's the problem: Here us formula 5.1.4: I'm not too sure where to start. Thanks!
5
votes
3answers
208 views

Using an identity to simplify the sum

So I ran into this problem today. It asks me to use an identity to simplify the sum. $$\sum_{j=7}^{27}\ln\left(\frac{j+1}{j}\right)$$ I have no idea where to start. I don't know any ...
0
votes
2answers
60 views

Sum of the first $n$ numbers that is neither divisible by 2 nor 3.

Show that the sum of the first $n$ positive integers that are divisible by neither 2 nor 3 is $\frac{3}{2}n^2-\frac{1}{2}$ if $n$ is odd and is $\frac{3}{2}n^2$ if $n$ is even. I have verified that ...
9
votes
2answers
136 views

Summation of series $\sum_{k=0}^\infty 2^k/\binom{2k+1}{k}$

How to find the sum of this series? $$\sum_{k=0}^{\infty}\cfrac{{2}^{k}}{\binom{2k+1}{k}}$$ It seems very easy. But I still can not work it out, can anyone help?
3
votes
1answer
54 views

An equality from Representation Theory

Studying Representation Theory of finite groups I've bumped in the following identity: $$\frac{n(n+1)}{2}=\sum_{i=1}^n\frac{(2i-1)!!(2n-2i+1)!!}{(2i-2)!!(2n-2i)!!}$$ My book suggests to prove it ...
1
vote
0answers
25 views

Pictorial derivation of sum of cubes

In the following picture, the formula for sum of squares of first $n$ natural numbers is derived using a clever construction of 3 triangles. This can be seen as a generalization to the legendary ...
0
votes
1answer
19 views

Finding roots for nested summations

Hi I was wondering how do I Solve this question. I have to solve for the root. I can solve for it when there's one summation but it's nested. I'm not that good at solving summations, if I can get some ...
2
votes
5answers
84 views

What is the sum of $\sum_{n=1}^\infty (n^2+n^3)x^{n-1}$?

Consider the power sequence $$\sum_{n=1}^\infty (n^2+n^3)x^{n-1}$$ What is the function to which it sums to? My reasoning is to differntiate the sum with respect to $x$, then to integrate with ...
1
vote
0answers
74 views

Very challenging series

Find a closed form for $\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^{2^2}}+\frac{1}{2^{2^{2^2}}}+\cdots $ Since I've never encountered this type of series before I was hoping someone here could help me ...
0
votes
4answers
63 views

Does the series $\sum \frac{1}{n\ (\ln(n))^{3/2}}$ converge or diverge?

Consider $$\sum \frac{1}{n\ \ln^{3/2}(n)}$$ The ratio test is inconclusive. The root test is inconclusive. And it seems right that $\frac{1}{n\ (\ln(n))^{3/2}}\leq\frac{1}{n}$ which diverges, but ...
2
votes
3answers
61 views

Show that $ \sum_{k=0}^{r} \binom{r-k}{m} \binom{s+k}{n} = \binom{r+s+1}{m+n+1} $?

I can't resolve this exercise and I need a tip. $$ \sum_{k=0}^{r} \binom{r-k}{m} \binom{s+k}{n} = \binom{r+s+1}{m+n+1} $$ where $ n \geq s $.