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

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-1
votes
0answers
40 views

The sum of $\frac{k^{2}}{k-1}$ from 6 to 12

Can you evaluate this sum by using the properties of the sigma notation ? Or I must develop this and evaluate them one by one ?
3
votes
3answers
64 views

Sum involving zeta functions

Find closed form of the following - $$ \displaystyle \sum_{n=2}^{\infty}{\left(\frac{(n-1)\zeta(n)}{4n-1}\right)} $$ I don't know how to approach to it - Using the integral definition? I cannot use ...
1
vote
2answers
133 views

Digit Sums: A Math Project

I'm doing a Math project on Digits Sums, and I don't really want to try to figure out all the 4-digit numbers that add up to 2, 3, 4, 5, and 6. That's partly because it would be hard, and also, ...
2
votes
2answers
78 views

If $(a+b)^n=\sum_{k=0}^{n}{n\choose k}a^{n-k}b^kc_k$, then $c_k=1$?

Be advised this is a real soft question: If $$(a+b)^n=\sum_{k=0}^{n}{n\choose k}a^{n-k}b^kc_k$$ Assuming $abc \neq 0$ must we have the following condition? $$c_k=1$$ for all $0 \leq k \leq n$ How do ...
2
votes
1answer
40 views

How many terms required in $e =\sum^∞_{k=0}{1\over k!}$ to give $e$ with an error of at most ${6\over 10}$ unit in the $20$th decimal place?

How many terms are required in the series $e =\sum^∞_{k=0}{1\over k!}$ to give $e$ with an error of at most ${6\over 10}$ unit in the $20$th decimal place? Here is what I have: $$e\approx ...
11
votes
5answers
474 views

Sums of rising factorial powers

Doodling in wolfram, I found that $$ \sum^{k}_{n=1}1=k $$ The formula is pretty obvious, but then you get $$ \sum^{k}_{n=1}n=\frac{k(k+1)}{2} $$ That is a very well known formula, but then it gets ...
0
votes
0answers
42 views

tricky derivative with logarithm of sum

I'm having trouble understanding the solution of a limit. It involves a formula for measuring certainty of a discrete probability distribution. Given a set of values $p_j$ which sum up to 1, find the ...
4
votes
1answer
129 views

Infinite Series $\sum_{n=2}^{\infty}\frac{(-1)^n}{n^k}\zeta(n)$

I can prove that $$\displaystyle \sum_{n=2}^{\infty}\frac{(-1)^n}{n}\zeta(n)=\gamma$$ Is there any known value for $\displaystyle \sum_{n=2}^{\infty}\frac{(-1)^n}{n^k}\zeta(n)$ for any natural number ...
1
vote
1answer
133 views

Why is $1+2+3+\cdots = 0 $? [duplicate]

I had seen this result a while back in a Numberphile video: $1+2+3+\cdots = -\frac{1}{12}$ I was trying to prove the same result using a different method when I accidently proved that the sum was ...
-5
votes
0answers
31 views

Summation from $2^0 log_2(n)$ to $2^{n-1} log_2(1)$ [on hold]

What is the sum: $2^0 log_2(n) + 2^1log_2(n-1) + 2^2log_2(n-2) + ... + 2^{n-2}log_2(2) + 2^{n-1}log_2(1)$?
-5
votes
6answers
57 views

What is $1+2+4+8+16+…+2^n$? [duplicate]

What is the result of: summation from one, two, four, eight until $n$ power of two? Thank you!
2
votes
2answers
52 views

How can I solve $\sum_{n=1}^{\infty} \frac{8}{(4n-1)(4n-3)}$ step by step? [on hold]

$$\sum_{n=1}^{\infty} \frac{8}{(4n-1)(4n-3)}$$ First off I know the answer to this question, but I do not know how to actually get the answer I do know some basic sums like that one. Can someone ...
1
vote
2answers
54 views

Find the Sum using bijection

Find the sum of $S=\displaystyle\sum_{i,j,k \ge 0, i+j+k=17} ijk$. I am looking for a solution that uses some bijection. I couldn't find any bijection. I am able to do the problem by other method by ...
1
vote
1answer
82 views

regularization of sum $n \ln(n)$

I was testing out a few summation using my previous descriped methodes when i found an error in my reasoning. I'm really hoping someone could help me out. The function which i was evaluating was ...
1
vote
4answers
37 views

Solving this Recurrence Relation in terms of previous values.

What will be the value of $X(n)$ and $Y(n)$ in terms of given $n,X(0),Y(0)$. $$ X(n) = X(n-1) + Y(n-1) \\ Y(n) = 2X(n-1) + Y(n-1) $$
0
votes
2answers
26 views

Simplifying with Summation

This is a problem out of my statistics book but my issue is simplifying from Step 3 to Step 4 below: Step 1: var X=$\sum\:p_i\:(x_i-E[X])^2$ Step 2: var X=$\sum\:p_i[x_i^2+E[X]^2-2x_iE[X]]$ Step ...
3
votes
1answer
37 views

bounding a sum using a definite integral

Conjecture. Let $1<p<\infty$. Then there exists $C\in(0,\infty)$ such that for any $k\in\mathbb{Z}^+$ we have \begin{equation}\sum_{n=1}^k(k+1-n)^{-\frac{p}{p+1}}n^{-\frac{1}{p+1}}\leq ...
6
votes
4answers
255 views

How to closed the sum $\displaystyle \sum_{k=0}^n \dfrac{(-1)^k(2k+1)!!}{(n-k)!k!(k+1)!}$

How to closed the sum $\displaystyle S=\sum_{k=0}^n \dfrac{(-1)^k(2k+1)!!}{(n-k)!k!(k+1)!}$ I'm trying divide two cases $n$ odd and $n$ even. I predict that ...
5
votes
2answers
154 views

What is the integer part of $\sum_{i=2}^{9999} \frac {1}{\sqrt i}?$

What is the integer part of: $$\sum_{i=2}^{9999} \frac {1}{\sqrt {i}}$$ A short but tricky problem. Any help is welcome.
1
vote
2answers
33 views

Question about summations with an unknown lower limit

I am unsure of how to proceed about finding the solution to this problem. $$\sum_{i=6}^8(\sum_{j=i}^8 (j+1)^2)$$ Obviously the last step is not to difficult, but the fact that the lower limit for the ...
0
votes
0answers
18 views

Does this theorem concerning upper and lower bnound of a monotone decreasing function have a formal name?

This is the theorem: Let $g$ be a monotone decreasing function and let $a,b \in \mathbb{N}$. Then the following holds true: $$\int_{a}^{b+1}g(x)dx \overset{(i)}{\leq} ...
12
votes
0answers
647 views
+100

Are there unique solutions for $n=\sum_{j=1}^{g(k)} a_j^k$?

Edward Waring, asks whether for every natural number $n$ there exists an associated positive integer s such that every natural number is the sum of at most $s$ $k$th powers of natural numbers ...
3
votes
3answers
51 views

Does $\lim\limits_{n \to +∞} \sum_{k=1}^n \frac{n\cdot \ln (k)}{n^2+k^2}$ diverge?

Does the limit of this summation diverge? $$\lim\limits_{n \to +∞} \sum_{k=1}^n \frac{n\cdot \ln (k)}{n^2+k^2}$$ Thanks!
1
vote
2answers
47 views

Solve $\frac{1}{2^\theta}\sum_{k=0}^{\theta} {\theta\choose k} \delta(k)=\theta$ for $\delta$

The following arises in unbiased estimation of a parameter for the binomial distribution, but that information is not needed for solving the question. I tried solving this by manipulating the sum to ...
2
votes
3answers
118 views

Calculate a multiple sum of inverse integers.

The question is to calculate a following sum: \begin{equation} {\mathcal S}_p(n) :=\sum\limits_{1\le j_1 < j_2 < \dots <j_p \le n-1} \prod\limits_{q=1}^p \frac{1}{n-j_q} \end{equation} for ...
8
votes
4answers
275 views

Prove that $\sum_{k=0}^{m}\binom{m}{k}\binom{n+k}{m}=\sum_{k=0}^{m}\binom{n}{k}\binom{m}{k}2^k$ [duplicate]

Prove that $$\sum_{k=0}^{m}\binom{m}{k}\binom{n+k}{m}=\sum_{k=0}^{m}\binom{n}{k}\binom{m}{k}2^k$$ What should I do for this equation? Should I focus on proving ...
4
votes
2answers
127 views

A summation involving multinomial coefficient

We need to find out $$\sum {\binom{N}{a_1,a_2,a_3...a_B} a_1^{\alpha}a_2^{\alpha}...a_C^{\alpha} }$$ $$a_1+a_2...a_B=N, \alpha>0 ,0\lt C \le B$$ All are nonnegative integers. We need to sum ...
2
votes
4answers
126 views

Prove that $1.49<\sum_{k=1}^{99}\frac{1}{k^2}<1.99$

It can be proven by induction that $$\sum_{k=1}^{n}\frac{1}{k^2}\leq2-\frac{1}{n}$$ From here, we can easily acquire the upper bound of the sum $$\sum_{k=1}^{99}\frac{1}{k^2}$$ letting $n=100$. ...
2
votes
4answers
137 views

Summing n times binomial(n,k)

I'm trying to do $\sum_{n=a}^b \left( \begin{array}{rl} n \\ a \end{array} \right) n $ . Is there a formula, that anybody knows?
3
votes
2answers
62 views

How is the Radius of Convergence of a Series determined?

Consider $$\sum_{n=0}^{\infty}\frac{(-1)^nx^n}{(n+1)^2}$$ which by the ratio test the ratio of two consecutive terms converges to $|x|$ as $n\rightarrow \infty$ and has a radius of convergence equal ...
0
votes
1answer
27 views

Combined pulling power of multiple engines in a train

Apologies if this question doesn't belong here.. Its very common to see multiple engines being used when there are more coaches in a train. In a configuration with more than one engine a good amount ...
-3
votes
1answer
34 views

What is the value of this sum? $\sum_{r=k}^{\infty} \frac{1}{r^{3/2} e^{\frac{c k}{2 r}}}$ [on hold]

How can I calculate this, where $c>0$? $$ \sum_{r=k}^{\infty} \frac{e^{\frac{-c k}{2 r}}}{r^{3/2}} $$
0
votes
1answer
15 views

How many ways are there to express a natural as a sum of 3 others—but by induction?

I have figured out an (inductive?) process, but I cannot express it formally: There is always one possibility where $n$ is in the first place of our 3-tuple: $[n~~0~~0]$. Then I can subtract $k~(\leq ...
4
votes
4answers
107 views

Combinatorial Proof for Binomial Identity: $\sum_{k = 0}^n \binom{k}{p} = \binom{n+1}{p+1}$ [duplicate]

I am studying combinatorics and I came across the identity $$\sum\limits_{k=0}^n \binom kp =\binom {n+1}{p+1}.$$ I have read the algebraic proof and it does not appeal to me. Is there an elegant ...
1
vote
3answers
73 views

Surprising Summation (3): $\sum_{i=1}^n\sum_{j=1}^i 2(n-i)+1=\sum_{i=1}^n i^2$

Show, without expanding the summation, that $$\sum_{i=1}^n\sum_{j=1}^i 2(n-i)+1=\sum_{i=1}^ni^2$$ It would be interesting to see different approaches to this problem, when expansion is not ...
2
votes
2answers
112 views

How prove this$\sum_{i=0}^{m-1}\binom{n-1+i}{i}x^ny^i+\sum_{j=0}^{n-1}\binom{m-1+j}{j}x^my^j=1$

let $m,n$ be positive numbers,and $x,y>0$ such $x+y=1$,show that $$\sum_{i=0}^{m-1}\binom{n-1+i}{i}x^ny^i+\sum_{j=0}^{n-1}\binom{m-1+j}{j}x^jy^m=1$$ My try: ...
0
votes
0answers
46 views

Binomial Sum Formula

I can't find a good closed form expression for this, $\sum_{k=0}^n\left[\binom{n}{k}\binom{m}{k}\right]$, where n is the variable, and m is a fixed constant, to be included in the formula. :( Can ...
4
votes
0answers
62 views

On finding an explicit form of a particular recurrence relation

Let $f$ be integrable over the interval $[0, 1]$, and $$I_n = \int_0^{1} x^n f(x) \, \mathrm{d}x.$$ Suppose $f(x) = f(1-x)$; we can then show that $$I_n = \sum_{k=0}^{n} \binom{n}{k} (-1)^k \, ...
13
votes
3answers
226 views

What is $\sum_{r=0}^n \frac{(-1)^r}{\binom{n}{r}}$?

Find a closed form expression for $$\sum_{r=0}^n \dfrac{(-1)^r}{\dbinom{n}{r}}$$ where $n$ is an even positive integer. I tried using binomial identities, but since the binomial ...
0
votes
3answers
103 views

Solve the recurrence $T(n) = 2T(n-1)+n^2$

Solve the recurrence $$T(1) = 1, T(2) = 1, T(3) = 1,T(n) = 2T(n-1)+n^2, n > 3$$ I have now, $$T(n) = 2T(n-1)+^2 $$ $$= 2(2T(n-2)+(n-1)^2+n^2$$ $$=4T(n-2)+2(n-1)^2+n^2$$ $$....$$ ...
2
votes
3answers
321 views

Double Summation: Need help to handle $ i \neq j $ : $ \sum_{i=0 \to 7,\ j=1 \to 8,\ i\neq j} (8i + j) $

[Q1]. Can I ? ( write the same summation as ) : $$ \sum_{i=0, i \neq j}^7 \sum_{j=1}^8 (8i + j) \tag{1}$$ I tried to solve the following Summation as follows: Let i = m-1 then, $ \sum_{i=0,\ i ...
6
votes
1answer
127 views

Does $(1^a+2^a+3^a+4^a+5^a)^b=1^c+2^c+3^c+4^c+5^c$ imply $(a,b,c)=(1,2,3)$?

Question : Is the following proposition true? Proposition : For positive integers $a,b,c$ where $b\ge 2$, if $$(1^a+2^a+3^a+4^a+5^a)^b=1^c+2^c+3^c+4^c+5^c$$then $(a,b,c)=(1,2,3)$. This is ...
3
votes
2answers
97 views

Sum of series ${n\choose 2a}{a\choose 0}+ {n\choose {2a+2}}{{a+1}\choose 1} + {n\choose {2a+4}}{{a+2}\choose 2} + \ldots$

I wanted to check the rationality of the cosine function for some rational multiples of $\pi$. And I found out that, $\cos(n \cdot\arccos x)$ generates a polynomial in $x$ whose co-efficients have the ...
4
votes
2answers
116 views

Finite sum related to Stirling numbers

I am wondering if there is a closed form solution for the following sum: $$ \sum _{k =0}^{n-1} \frac{(-1)^{k} (n-k)^{n+1} }{(k+1)(k+2)}\binom{n}{k}. $$ If the the factors $(k+1)(k+2)$ in the ...
13
votes
6answers
494 views

Combinatorial interpretation of an alternating binomial sum

Let $n$ be a fixed natural number. I have reason to believe that $$\sum_{i=k}^n (-1)^{i-k} \binom{i}{k} \binom{n+1}{i+1}=1$$ for all $0\leq k \leq n.$ However I can not prove this. Any method to prove ...
1
vote
3answers
72 views

Missing steps: Show the sum of the first n positive integers is of order $n^2$

In Rowsen's Discrete Mathematics text, 6th edition. He has this problem as an example (#11) on page 190. His solution for obtaining a lower bound is to ignore the first half of the terms. He does the ...
-1
votes
1answer
69 views

Simplifying a sum of products related to Vandermonde determinant

How to show this equality? $$ 1=(-1)^n\sum_{k=0}^n\frac{x_k^n}{\prod_{\substack{l=0 \\ l \neq k}}^n(x_l-x_k)} $$ This is part of a proof to show the value of the determinant of the Vandermonde matrix ...
0
votes
2answers
48 views

find the coefficient

If $n$ is an odd natural number, and $\sin(n\theta) = \Sigma_{r=0}^{n} b_r \sin^r\theta$, then find $b_r$ in terms of $n$. I have tried this using trigonometric expansion but unable to find solution ...
3
votes
3answers
93 views

Upper and lower bounds for $S(n) = \sum_{i=1}^{2^{n}-1} \frac{1}{i} = 1+\frac{1}{2}+ \cdots +\frac{1}{2^n-1}.$ [duplicate]

For a positive integer $n$ let $S(n) = \sum_{i=1}^{2^{n}-1} \frac 1i = 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+ \cdots +\frac{1}{2^n-1}.$ Then which of the following are true. (a) ...
1
vote
2answers
85 views

Solving $x^{2n} = \frac{1}{2^n}$

What is the principle behind solving for a variable that is raised to another variable? I came across this problem doing infinite sums: I had to solve the equation $$x^{2n} = \frac{1}{2^n}$$ for ...