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

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1
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6answers
87 views

Prove by induction $3+3 \cdot 5+ \cdots +3 \cdot 5^n = \frac{3(5^{n+1} -1)}{4}$

My question is: Prove by induction that $$3+3 \cdot 5+ 3 \cdot 5^2+ \cdots +3 \cdot 5^n = \frac{3(5^{n+1} -1)}{4}$$ whenever $n$ is a nonnegative integer. I'm stuck at the basis step. If I ...
0
votes
0answers
25 views

Fubini's theorem (interchange of sum and integrals) in case of multivariable function

Can the Fubini's theorem in case of single variable sequence of functions be readily extended to multivariable sequence of functions?, i.e, Is it true to say $$\iiint_V\sum_{n=0}^\infty f_n(u,v,w) \,...
1
vote
1answer
26 views

Isolate x in the equation

I'm trying to isolate $x$ from the next equation. Some ideas? \begin{equation} k_{1} = k_{2}\sum_{i=1}^{x}\binom{x-1}{i-1}\frac{1}{i^{k_{3}}}, \end{equation} where $k_{1}$, $k_{2}$ and $k_{3}$ are ...
3
votes
4answers
42 views

Evaluate $\sum^{n}_{i=1}(10^{i+1}-10^i)$

Evaluate $$\sum^{n}_{i=1}(10^{i+1}-10^i)$$ Here's what I did $$\sum^{n}_{i=1}(10^{i+1}-10^i) \\ = 10(\sum^n_{i=1} 1^{i+1}-1^i) \\ = -10(\sum^n_{i=1} 1^i-1^{i+1}) \\= -10(1^n-1^{n+1}) \\= 10^{n+1}-...
7
votes
1answer
43 views

Numbers on a circle: how many arc sums can be positive?

There are $n$ real numbers, $a_1,\dots,a_n$, arranged on a circle. Given a fixed integer $k<n$, let $S_i$ be the sum of the $k$ adjacent numbers starting at $a_i$ and counting clockwise, like this (...
8
votes
3answers
254 views

Asymptotic behaviour of sum over the inverse japanese symbol

I am interested in the asymptotic behavior of the sum $$\sum_{m=1}^M\frac{1}{\sqrt{m^2+\omega}}$$ for $1>\omega>0$ in the Limit $M\to\infty$ up to order $\mathcal{O}(M^{-1})$. The first thing I ...
3
votes
1answer
508 views

Sum of floor of harmonic progression: $\sum_{i=1}^n\lfloor\frac ni\rfloor=2\sum_{i=1}^k\lfloor\frac ni\rfloor-k^2$ for $k=\lfloor\sqrt n\rfloor$

This question is actually from a programming question that a formula is required to compute maths faster. (Please note that computer frequently rounds down to the nearest integer, thus the floor ...
2
votes
1answer
46 views

Summation involving a hypergeometric 1F1 function

I'm trying to find a closed form for the following: \begin{equation} \sum_{n=0}^\infty \frac{(-1/4)_n}{n!(3/2)_n}\left(\frac{i}{2\tau}\right)^{n} {_1F_1(2n+1;2n+2;i k)} \end{equation} Using the ...
2
votes
4answers
6k views

Proof by induction that $ \sum_{i=1}^n 3i-2 = \frac{n(3n-1)}{2} $

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: $$ \sum_{i=1}^...
2
votes
3answers
53 views

Evaluate $\sum_{i=1}^{25} 2i(i-1)$

Evaluate the sum: $$\sum_{i=1}^{25} 2i(i-1)$$ All I could do is: $$2 \sum_{i=1}^{25} i (i-1)$$ What can I do after this? Is there a way to evaluate without inserting every single integers? ...
3
votes
4answers
137 views

Prove if $n \in \mathbb N$, then $\frac{1}{2!}+\cdots+\frac{n}{(n+1)!}=1-\frac{1}{(n+1)!}$

Prove if $n \in \mathbb N$, then $\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+\cdots+\frac{n}{\left(n+1\right)!} = 1-\frac{1}{\left(n+1\right)!}$ So I proved the base case where $n=1$ and got $\frac{1}{2}...
6
votes
3answers
677 views

Prove this using counting techniques: $\sum_{k=0}^{n}{\binom{2n+1}k} = 2^{2n}$

I recently came across a question while studying for an exam. I haven't been able to solve it. We had to prove: $$\sum_{k=0}^{n}{2n+1\choose k} = 2^{2n}$$ We had to use counting techniques. This was ...
3
votes
3answers
45 views

Summation of series involving factorials.

I got this question in a maths contest archive and I am completely clueless over how to start. $$\sum_{m=0}^q(n-m){(p+m)!\over m!}= {(p+q+1)! \over q!}\left(\frac{ n}{ p+1}-\frac {q}{p+2}\right)$$ I ...
0
votes
2answers
59 views

Transform double sum $\sum_{i=0}^\infty \sum_{j=0}^i$ to $\sum_{i=0}^\infty \sum_{j=0}^\infty$?

Consider a double sum (assuming it converges) $$\sum_{i=0}^\infty \sum_{j=0}^i f(i,j)$$ Is there a convenient way to rewrite this sum so that both summations go from zero to infinity $\sum_{i=0}^\...
3
votes
0answers
131 views

Finding sum for function with floor-function

I am trying to find a formula to calculate the following sum: $$\sum_{x=0}^n (2ax - {1 \over 2}a^2 - {1 \over 2} a) $$ where $$ a = \left\lfloor {x \over \phi^2} \right\rfloor $$ and $$\phi = {1 + ...
2
votes
2answers
32 views

Question about simplification in summation

I am reading a book where the following example is shown. $$= \sum_{0\le n-j\le n} (a+b(n-j)) $$ $$= \sum_{0\le j\le n} (a+bn-bj) $$ Why is n-j being simplified to j? I don't understand why this is ...
1
vote
1answer
45 views

How can I simplify: $\sum^{n-1}_{i=1}\sum^{n}_{j=i+1}\sum^{j}_{k=1} 1?$

I simplified the most inner sum to: $j$. So, now I have: $$\sum^{n-1}_{i=1}\sum^{n}_{j=i+1}j$$ I'm not sure if the following is correct but here is what I am thinking. I can re-index the inner sum ...
0
votes
0answers
22 views

Another nasty multiple sum.

Let $s \ge 0$ be an integer, let $a_\xi \ge 1$ for $\xi=0,\cdots,s$ and let $t$ and $\beta$ be parameters. Also defines $l:= a_0+\cdots+a_s$. Consider a following multiple sum: \begin{equation} {\...
2
votes
4answers
106 views

increasing sum of binomial coefficients

I've been working on a problem and got to a point where I need the closed form of $$\sum_{k=1}^nk\binom{m+k}{m+1}.$$ I wasn't making any headway so I figured I would see what Wolfram Alpha could do....
0
votes
3answers
80 views

Convergence problem $\sum \left(1-n\sin\left(\frac{1}{n}\right)\right)$ [on hold]

I have to check convergence of: $$\sum_{n=1}^\infty\left(1-n\sin\left(\frac{1}{n}\right)\right).$$ I have no idea but I only check that $\lim \ n\left(1-n\sin\left(\frac{1}{n}\right)\right)=0$.
0
votes
2answers
61 views

Sum involving $\cosh$ and $\sinh$

I would like to prove the equation $$\frac{\sinh\left(\left (1-\frac{1}{2m} \right)x\right)}{\sinh(x/2m)}=1+ \sum\limits_{n=1}^{m-1}2\cdot \cosh\left(\left( 1-n/m \right)x\right),\quad \forall x > ...
6
votes
3answers
133 views

Proving $\sum\limits_{k=0}^n \sum\limits_{j=0}^{n-k} \frac{(k-1)^2}{k!} \frac{(-1)^j}{j!} =1$ without character theory

Let $n \geq 2$ be an integer. I would like to prove the following identity in an easy way: $$\sum\limits_{k=0}^n \left( \frac{(k-1)^2}{k!} \sum\limits_{j=0}^{n-k} \frac{(-1)^j}{j!} \right)=1$$ You ...
42
votes
4answers
8k views

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{...
4
votes
3answers
572 views

Give the combinatorial proof of the identity $\sum_{i=0}^{n} \binom{k-1+i}{k-1} = \binom{n+k}{k}$

Given the identity $$\sum_{i=0}^{n} \binom{k-1+i}{k-1} = \binom{n+k}{k}$$ Need to give a combinatorial proof a) in terms of subsets b) by interpreting the parts in terms of compositions of ...
2
votes
5answers
295 views

$33^{33}$ is the sum of $33$ consecutive odd numbers. Which one is the largest? (Q25 from AMC 2012)

The number $33^{33}$ can be expressed as the sum of $33$ consecutive odd numbers. The largest of these odd numbers is $\mathrm{A.}\ 33^{32} +32$ $\mathrm{B.}\ 33^{31} +32$ $\mathrm{C.}\...
1
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1answer
39 views

Alternative proofs that Dirichlet products are associative?

Is there alternative proof of the following fact: Dirichlet product on arithmetic function is associative. I'm looking for something different than that given in Dirichlet's product with ...
0
votes
0answers
28 views

The sum $\sum_{i=1}^n \frac{x^i}{\Gamma(\frac{i}{\sqrt{n}})}$? [on hold]

I'm interested in evaluating the following sum : $\displaystyle{\sum_{i=1}^n \frac{x^i}{\Gamma(\frac{i}{\sqrt{n}})}}$ where $x>0$. The existence of a closed form would be great but is perhaps too ...
0
votes
1answer
41 views

Kolmogorov's Truncation Lemma (iii)

Probability with Martingales: In the definition of $f$, is that really $z$ and not $\lceil |z| \rceil$, $\lfloor |z| \rfloor + 1$ or something? How exactly do we have the part in the $\...
8
votes
2answers
260 views

Evaluate $ \int_{0}^{1} \log\left(\frac{x^2-2x-4}{x^2+2x-4}\right) \frac{\mathrm{d}x}{\sqrt{1-x^2}} $

Evaluate : $$ \int_{0}^{1} \log\left(\dfrac{x^2-2x-4}{x^2+2x-4}\right) \dfrac{\mathrm{d}x}{\sqrt{1-x^2}} $$ Introduction : I have a friend on another math platform who proposed a ...
-1
votes
2answers
86 views

How can I calculate $\sum_{i=0}^{n}\sum_{j=i}^{n} \binom{n+1}{j+1}\binom{n}{i}$

I tried Google and various ways, including walking the list questions in the chronological order as far. How can I show $$\sum_{i=0}^{n}\sum_{j=i}^{n} \binom{n+1}{j+1}\binom{n}{i}=2^{2n}$$ ...
5
votes
1answer
187 views

Is there any closed form for $\sum_{k=1}^n \frac{1}{k^k}$?

Is there any closed form for the summation: $$\sum_{k=1}^n \frac{1}{k^k} = ? $$ or at least a tight lower bound?
1
vote
3answers
118 views

Evaluation of $\sum_{n=1}^{\infty} \frac {x^{-n}}{n}$

I wanted to evaluate $$ \sum_{n=1}^{\infty} \frac {2^{-n}}{n} $$ And noticed that for any base it has a pattern, so now I want to know how to evaluate $$ \sum_{n=1}^{\infty} \frac {x^{-n}}{n} $$ I ...
62
votes
27answers
19k views

Prove 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 ...
2
votes
2answers
76 views

Calculate two sums: $\sum_{i=1}^{99}\frac{1}{\sqrt{i+1}+\sqrt{i}}$, $\sum_{i=1}^{9999}\frac{1}{(\sqrt{i}+\sqrt{i+1}) (\sqrt[4]{i}+\sqrt[4]{i+1})}$.

Calculate $$\sum_{i=1}^{99}\frac{1}{\sqrt{i+1}+\sqrt{i}}$$ I've figured out that the answer is 9 -there is a pattern that I've figured out. I've created a code and solved it... but how could I do it ...
2
votes
3answers
48 views

induction clarification about the step $n+1$

Suppose i need to prove that $\frac{1}{2^2}+\frac{1}{3^2}...+\frac{1}{n^2}<1-\frac{1}{n}$ So in the step of $n+1$, the right side becomes $<1-\frac{1}{n+1}$ or is it: $<1-\frac{1}{n}-\frac{1}...
0
votes
1answer
30 views

Sum algebra solving for coefficient

Is the following equation solvable for $k$? $$\sum_{i=1}^{n}\frac{x_ie^{kx_i}}{1+e^{kx_i}} = 0$$
0
votes
2answers
18 views

Interval for summation function?

I know that you almost always set domains for a summation function $\left( \sum \right)$, but can you also set an interval for that domain? Say the domain was 1 to 10, could I set an increment of 0.5 ...
2
votes
3answers
84 views

Formula for $\sum_{i\geq 0} i{n \choose 2i}$?

So I know that $\sum_{i\geq 0}{n \choose 2i}=2^{n-1}=\sum_{i\geq 0}{n \choose 2i-1}$. However, I need formulas for $\sum_{i\geq 0}i{n \choose 2i}$ and $\sum_{i\geq 0}i{n \choose 2i-1}$. Can anyone ...
2
votes
0answers
50 views

Simplifying a Double Summation

Let $f_n(k)$ be defined as $$f_n(k)=\sum_{i=1}^n\sum_{j=1}^i\left(\frac{j}{i}\right)^k$$ Can $f_n(k)$ be simplifying down to an expression without summations? By simply graphing $f_n(k)$, it is clear ...
0
votes
2answers
48 views

Finding $\lim_{L \to \infty} \exp{\frac{T}{L}}\sum_{i=1}^L[ \exp{iA + (i-1)B}]$

I am working on a problem and I have come up with a formula that I would like to simply. WLOG, it looks like the following: $\exp{\frac{T}{L}}\sum_{i=1}^L[ \exp{iA + (i-1)B}]$ Here, $A,B, T$ are ...
1
vote
5answers
109 views

The solution of equation $4+6+8+10+\cdots +x =270$ is 15. [closed]

The solution of equation $4+6+8+10+\cdots +x =270$ is $x=15$. How can I prove it? I ve tried the geometric sequence but I cannot figure out the pattern.
3
votes
1answer
57 views

Sum of $x_1^k+x_2^k+\dots+x_n^k$

I was recently wondering if there is some quicker way to compute $x_1^k+x_2^k+\dots+x_n^k$ for any natural $k$ than just exponentiation and adding one-by-one? Thanks in advance.
1
vote
1answer
42 views

Summation of factorial.

$$2(\frac{1}{3!\times7!}+\frac{1}{1\times9!})+\frac{1}{5!\times5!}=\frac{2^a}{b!}$$ find $a,b$ by some predictions I see $b=10$ but what about numerator. I think we have to $\sum {N\choose r}=2^N$ but ...
2
votes
0answers
41 views

sum of subset of complex numbers [duplicate]

let there be $\{z_1 ,..., z_n\}$ a group of complex numbers. Show that there's a subset $J \subset \{1,...n\}$ so that $$\lvert \sum_{k \in J}z_k \rvert \ge \frac{1}{4\sqrt2}\sum_{i=1}^n\lvert z_i \...
0
votes
0answers
60 views

Does there exists a closed expression for the following sum? [closed]

Does there exists a closed expression for the following sum? $\sum_{i = 1}^n i {m \choose i}$
1
vote
1answer
44 views

Generalizing a Telescoping Sum $\sum_{n=1}^\infty \frac{1}{n+k}-\frac{1}{n}$

I was trying to generalize an integral I found yesterday on this website and ran into the following interesting sum: $S_k=\sum_{n=1}^\infty \frac{1}{n+k}-\frac{1}{n}$. I have seen this sum come up a ...
0
votes
2answers
61 views

the sum of all four digit multiples of 6

The sum of all four digit multiples of $6$ is equal to: A. $8~274~489$ B. $8~247~498$ C. $8~241~996$ Can you help me with this question? I've tried $$S_n= \frac{n(a_1+a_n)}{2}$$ with $...
0
votes
1answer
41 views

How to find the General expression of $\sum_{k=0}^ {\lfloor n/3\rfloor} {n \choose 3k}$ [duplicate]

Well as the title says I'm having problems trying to derive a general expression for this sum which involves cubic roots of unity $$\sum_{k=0}^ {\lfloor \frac n 3\rfloor} {n \choose 3k}$$ Need help ...