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
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
36 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
39 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
26 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
99 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
22 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
36 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
24 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
386 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 [on hold]

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
37 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
58 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
207 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
133 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
73 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 $.
3
votes
4answers
185 views

What is $\lim_{n\to \infty}\sum_{k=1}^n \left(\frac{k}{n}\right)^n$?

I'm asked to find $\displaystyle\lim_{n\to \infty}\sum_{k=1}^n \left(\frac{k}{n}\right)^n$. It seems that the sum converges. I managed to prove that $\forall n,1<\sum_{k=1}^n ...
-1
votes
5answers
102 views

What is $\sum_{n=1}^\infty\frac{1}{4^n}=?$ [on hold]

How do I calculate this sum: $$\sum_{n=1}^\infty\frac{1}{4^n} \text{?}$$ I know how to show that the series converges but don't how to achieve its value.
0
votes
2answers
19 views

summation of polynomial products

I need help in understanding how the summation of the product of two polynomials is written. $(a_{0} +a_{1}x +a_{2}x^{2})(b_{0} +b_{1}x + b_{2}x^{2}) =\\ (a_{0}b_{0})x^{0} + (a_{1}b_{0} + ...
-1
votes
0answers
26 views

Summation of a series involving modulus. [closed]

how to evaluate this summation value ..?? Σ((i^3)*(N % i)) for i(1 to N) I think there is some formula which comes up to solve this summation.. ??
0
votes
1answer
30 views

closed form for $\sum_{k=1}^n\left[ 2^k\binom{2n-k}{n-k}-2^{k+1}\binom{2n-k-1}{n-k-1}\right ]k^s$

$$\sum_{k=1}^n\left[ 2^k\binom{2n-k}{n-k}-2^{k+1}\binom{2n-k-1}{n-k-1}\right ]k^s$$ by some steps I got $$\sum_{k=1}^n\left[ 2^k\binom{2n-k}{n-k}-2^{k+1}\binom{2n-k-1}{n-k-1}\right ...
3
votes
2answers
32 views

find $\sum_{k=0}^{t}(-1)^k\binom{t}{k}^2$ for odd t then for even t

find $$\sum_{k=0}^{t}(-1)^k\binom{t}{k}^2$$ for $t=2n$ then for $t=2n+1$ I tried by expand $(1-x)^n(1-x)^n$, with no result. Any Help ?
1
vote
3answers
55 views

Find $a_1$ given that $(1+x)^{100} = \sum_{i=0}^{100} a_ix^i$

If $(1+x)^{100} = \sum_{i=0}^{100} a_ix^i$, then $a_1$ is .. The options are $1$, $2$, $99$ or $100$. I'm sure the problem is trivial, but I just don't understand what is meant.
2
votes
1answer
53 views

How can I compute the following fast?

What approach should I adopt for computing the following problem fast? $$f(n) = \sum_{i=1}^n (n \mod i)$$ $$ 1\le n \le 10^{10}$$ Since the answer can be huge I have to output it modulo some given ...
1
vote
2answers
44 views

Closed form sum for the series given below?

Does the following series have a closed form sum? $$f(n,r) = \sum_{i=0}^n \binom{r+i}{r}$$
2
votes
2answers
29 views

Summation of series

Find $\sum_1^n$ $\frac {2r+1}{r^2(r+1)^2}$ Also, find the sum to infinity of the series. I tried decomposing it into partial fractions of the form $\frac Ar$ + $\frac{B}{r^2}$ + $\frac{C}{(r+1)}$ + ...