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

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3
votes
2answers
54 views

Finding Exact Values of Specific Infinite Series

Prove that $\Sigma_{n=1}^{\infty}(n/2^n)=2$ and that $\Sigma_{n=1}^{\infty}(n^2/2^n)=6$. Thoughts: I have a feeling that if someone shows me how to do one, I'll be able to figure out the other. So ...
0
votes
0answers
21 views

How to test the convergence of the following sereis?

$\sum^{\infty}_{n=1}\sin{\frac{1}{n}}$: The only test I can think of for this one is basic comparison ($\sin{\frac{1}{n}}\le\frac{1}{n}$). But $\frac{1}{n}$ diverges. ...
1
vote
1answer
51 views

Summing $n$ numbers so that they equal $0 \mod{n}$

Let $A_n=\{(a_1,a_2,\dots,a_n) :\sum_{i=1}^na_i=0\mod{n}\}$, where $a_i\in[n-1]$. How many elements are in $A_n$? My initial attempt was a stars-and-bars argument. For example, let $n=4$. Then we ...
2
votes
2answers
30 views

How to test the convergence of $\sum^{\infty}_{n=1} \frac{3^n(n-1)(-1)^n}{n^3}$?

So $\sum^{\infty}_{n=1} \frac{3^n(n-1)(-1)^n}{n^3}=\sum^{\infty}_{n=1} \frac{(-3)^n(n-1)}{n^3}$. Then I fail to continue. Is there a way to simplify it?
2
votes
1answer
24 views

How to compute the following series using taylor expansion manipulation?

How to compute $\sum^{\infty}_{n=0} \frac{x^n}{(n+2)n!}$ and $\sum^{\infty}_{n=0}(-1)^n \frac{(n+1)x^{2n+1}}{(2n+1)!}$ using taylor expansion manipulation? $1.\sum^{\infty}_{n=0} ...
2
votes
1answer
18 views

How to compute taylor series for $f(x)=\frac{1}{1-x}$ about $a=3$ and $f(x)=\sin{x}$ about $a=\frac{\pi}{4}$?

How to compute taylor series for $f(x)=\frac{1}{1-x}$ about $a=3$ and $f(x)=\sin{x}$ about $a=\frac{\pi}{4}$ ? For the first one, using substitution, let $t=x-3$, then $x=t+3$. Then ...
1
vote
2answers
34 views

How to test the convergence of $\sum^{\infty}_{n=2}\frac{1}{\ln{n}}$ and $\sum^{\infty}_{n=2}\frac{\ln{n}}{n^{1.1}}$?

How to test the convergence of $\sum^{\infty}_{n=2}\frac{1}{\ln{n}}$ and $\sum^{\infty}_{n=2}\frac{\ln{n}}{n^{1.1}}$? For the first one, I use basic comparison and compare it to $\frac{1}{n}$, since ...
0
votes
2answers
38 views

p-Norm with p $\to$ infinity

I have to show that: for all vectors $v\in \Bbb R^n$: $\lim_{p\to \infty}||v||_p = max_{1\le i \le n}|v_i|$ with the $||.||_p$ defined as $$ ||.||_p: (v_1, \dots ,v_n) \to (\sum^n_{i=i} ...
0
votes
0answers
26 views

Simplify/expand $\ln \left(\sum^n_{i=1}x_i^{\theta-1}\right)$

Can someone help simplify/expand this natural log? I want to bring the $\theta - 1$ down in front of the $\ln$ but I don't know how the rules work with the summation. $$\ln\left( ...
3
votes
1answer
49 views

How to do this infinite sum?

$$ F_\alpha(z)=\frac{2}{\pi \alpha}\sum_{n=0}^\infty \frac{1}{n^2+z/\alpha^2} $$ The answer is: $$ F_\alpha(z)=z^{-\frac{1}{2}}\coth(\pi z^{\frac{1}{2}}/\alpha)+\alpha/\pi z $$ It is easy to show ...
0
votes
2answers
42 views

Combinatorial identity involving binomial coefficients.

In order to conclude a proof (see last equality in B. Poonen's article), I need to establish the following identity: $$\forall (\ell,n)\in\mathbb{N}^2,\ell\leqslant n,\sum_{m=\ell}^n{n\choose ...
0
votes
2answers
29 views

How to come up with Gauss arithmetic progression solution in this sum

I need to solve this: $\sum\limits_{k=0}^n k\\$ Using this specific method: $n + \sum\limits_{k=0}^{n-1} k\ = 0 + \sum\limits_{k=1}^n k\\$ Now this has to evaluate to: $\ n+\frac{n(n-1)}{2} = ...
-1
votes
1answer
28 views

If p is an odd prime, compute $S = \sum^{p-1}_{k=1} (\frac{k}{p}) $ and $P = \prod^{p-1}_{k=1} (\frac{k}{p})$… [closed]

If $p$ is an odd prime, compute $S = \sum^{p-1}_{k=1} (\frac{k}{p}) $ and $P = \prod^{p-1}_{k=1} (\frac{k}{p})$, the sum and product of all nonzero Legendre symbols modulo p. Honestly have no idea ...
1
vote
4answers
78 views

Find the sum to $n$ terms of the series $10+84+734+…$

Find the sum to n terms of the series $10+84+734+....$ $\frac{9(9^n+1)}{10} + 1$ $\frac{9(9^n-1)}{8} + 1 $ $\frac{9(9^n-1)}{8} + n $ $ \frac{9(9^n-1)}{8} + n^2$ My attempt: I'm getting option ...
0
votes
1answer
77 views

Is there a closed form for $\sum_{k=0}^n \frac{x^k}{k!}$? [closed]

What is the closed form of $$\sum_{k=0}^n \frac{x^k}{k!}$$ as a function of $x$ and $n$? Knowing that it converges to $e^x$ when $n\to \infty$.
0
votes
1answer
31 views

Of Balls in Bins in Different Sections with Caps

Problem: There are $19$ bins: $7, 5, 7$ in the left, centre and right sections respectively. There are $8$ balls, some or all of which are to be put into these bins with the following ...
0
votes
1answer
44 views

Summation notation index n-1 refers to non-existent element? [closed]

I don't have any math background, and I'm trying to type this equation into R, and having difficulties understanding the summation notation. I dont understand the part, because it seems to refer ...
14
votes
1answer
114 views

Are there some techniques which can be used to show that a sum “does not have a closed form”?

I am aware that there are some techniques which can be used to show that some function does not have an antiderivative expressible using elementary functions, such as Liouville's theorem. (More ...
4
votes
4answers
108 views

Showing $\sum_{i = 1}^n\frac1{i(i+1)} = 1-\frac1{n+1}$ withoug induction?

I oversaw a high-school mathematics test the other day, and one of the problems was the following Show, using induction or other means, that $$\sum_{i = 1}^n\frac1{i(i+1)} = 1-\frac1{n+1}$$ The ...
0
votes
0answers
21 views

Limit of a summation where one part is increasing and the other decreasing- can't tell which way it's going

I'm doing a graph theory problem where given a tree $T$, $f(T)$ is the proportion of vertices which are leaves- so for the simplest case - a path through all $n$ vertices of $T$- there are two leaves ...
-4
votes
0answers
29 views

How can this summation equation be proved by mathematical induction? [duplicate]

How can I prove this by mathematical induction? Sorry the wording is not very spacious, but it says "for all integers n." $$\sum_{j=1}^{n}{j^3}= \left(\frac{n(n+1)}2\right)^{2} \text{ for all ...
1
vote
0answers
29 views

Sum of alternating sign groups of integer squares

This is a generalization of the question here, assuming the sign changes for groups of $k$ consecutive squares. What is the closed form for the summation? ...
3
votes
1answer
105 views

Find the sum of $1^{n}-2^{n}+3^{n}-4^{n}+\cdots+m^{n}$

After seing this question I started wondering about a generalization of a similar sum. The sum is $$ S(m,n)=\sum_{r=1}^{m}(-1)^{r-1}\;r^{n} $$ I gave this to WA to crunch and it gave $$ S(m,n)= ...
1
vote
1answer
22 views

Digit-sum division check in base-$n$

Several years ago now I realised that for any natural numbers $x$ and $y$ you could write $$x^y=(x-1) \left(\sum_{i=0}^{y-1}x^i\right)+1$$ This shows that $x^y-1$ will always be divisible by $x-1$, ...
9
votes
1answer
311 views

How to estimate a specific infinite sum

Let $M$ be an $n$ by $n$ matrix with each diagonal element equal to $k$ and each non-diagonal element equal to $k-1$ where $n$ and $k$ are positive integers. Let $k < n$ and we can assume both $k$ ...
1
vote
1answer
31 views

Summation with running indices that have running indices

I would like to be able to simplify an expression such as $$\prod_{l=1}^{i} \sum_{j_l=j_{l-1}+1}^{m-i+l-1} f(j_l),$$ with $j_o:=0$. For instance for $i=3,m=4$ the result is $f(1)f(2)f(3)$ whereas for ...
0
votes
0answers
18 views

What effect does sampling time have on a Fourier Series sum?

What effect would the sampling time of this Fourier sum have on it's accuracy? Is this to do with Nyquists theorem? or am I heading in the wrong direction with this question? Cheers
1
vote
1answer
68 views

Largest possible value of a sum

This question has been confusing me and I would love some help. If $M$ is $n$ by $n$, symmetric, positive definite and integer valued and $n$ is a fixed positive integer, what is the large possible ...
2
votes
1answer
48 views

Why are these sums equal?

I've been looking at some pretty cool proofs of $\zeta(2)=\frac{\pi^2}{6}.$ recently, and the one proof that was the easiest to understand for me was how Euler originally presented it, by finding and ...
3
votes
2answers
167 views

What is the infinite sum

$$S = {\frac{2}{5} \\+ \frac{2}{5}\cdot\frac{3}{7} \\+ \frac{2}{5}\cdot\frac{3}{7}\cdot\frac{5}{11} \\+ \frac{2}{5}\cdot\frac{3}{7}\cdot\frac{5}{11}\cdot\frac{9}{13} \\+ ...
1
vote
2answers
53 views

proving convergent sum inequality

I want to prove that for $m→∞ ⇒((1+x/m)^m→exp(x))$ My idea is to prove that there is an $m≥n$ so that $$(1+x/m)^m≥\sum_{j=0}^{n}x^j/j!≥(1+x/n)^n$$ now I would use the binomial theorem to rewrite ...
1
vote
2answers
58 views

Ratio Inequality

How can I prove that, $$\frac{a_{1}+a_{2}+\dots+a_{n}}{b_{1}+b_{2}+\dots+b_{n}} \le \max_i\left\{\frac{a_{i}}{b_{i}}\right\}$$ where $1 \le i \le n$, and $a_{i} \neq a_{j}$ and $b_{i} \neq b_{j}, ...
0
votes
0answers
31 views

How to find the following sums.

Let $$a=\sum _{r=1}^{11}tan^2\bigg(\frac{r\pi}{24}\bigg) $$and $$b=\sum _{r=1}^{11}(-1)^{r-1}tan^2\bigg(\frac{r\pi}{24}\bigg) $$ then find the value of $log_{(2b-a)}{(2a-b)}$. In $a$ I find that ...
-4
votes
2answers
56 views

Tricky summation of the series. [closed]

What does $$-50^2-2\cdot (49)^2-3\cdot(48)^2-.....-0\cdot0^2+1^2(51)+2^2(52)+...(50)^2100$$equal? I don't know what to do. Also I couldn't find any symmetry here so I am stuck over this problem. ...
0
votes
1answer
24 views

Sum of $k^2*x^k$

Been trying to figure out how to sum $\sum_{k=0}^{\infty} k^{2}x^{k}$ for x<1. I tried the trick through differentiation, but cannot include (k-1) to have second derivative of x. I googled it can ...
7
votes
6answers
178 views

Find the sum of $-1^2-2^2+3^2+4^2-5^2-6^2+\cdots$

Find the sum of $$\sum_{k=1}^{4n}(-1)^{\frac{k(k+1)}{2}}k^2$$ By expanding the given summation, $$\sum_{k=1}^{4n}(-1)^{\frac{k(k+1)}{2}}k^2=-1^2-2^2+3^2+4^2-5^2-6^2+\cdots+(4n-1)^2+(4n)^2$$ ...
3
votes
3answers
83 views

Bound on the infinite sum of logarithms

Is it possible to show that $X=\frac 12 \log 3 + \frac 14 \log 4 + \frac 18 \log 5 + \frac{1}{16} \log 6 + \dots < \log 4$? I think we can do $\frac 12 X= \frac 14 \log 3 + \frac 18 \log 4 + ...
0
votes
1answer
19 views

Combinatory sum of multiplications

Suppose i have $N$ variable. In a sum, i have terms each consist of combination of n variable. Each variable(they appear only once in one term) is to be multiplied to get term. How can i write the sum ...
3
votes
0answers
170 views
+100

Is there a way to write this recurrence relation in faster-to-program manner?

I have the following recurrence relation for some coefficients $$b_{n+2} = \frac{1}{(n+3)(n+2)P_0} \sum_{k=1}^n (n-k+2) (n-k+1) b_k b_{n-k+2}, \quad n>1$$ with $b_1$ to $b_3$ and $P_0$ being the ...
1
vote
1answer
27 views

Can this be done without substitution of values?

If $${ \left( 1-{ x }^{ 3 } \right) }^{ n }=\sum _{ r=0 }^{ n }{ { a }_{ r }{ x }^{ r }{ \left( 1-x \right) }^{ 3n-2r } } $$ then find $a_r$. My first attempt: I wrote the above equation as: ...
0
votes
0answers
13 views

Partial Fourier Series Error

I calculated the coefficient of the complex fourier series of a trapezoidal signal to be: $$c_n = \frac{\tau}{T} \frac{sin(0.5n\omega_0\tau) \cdot sin(0.5n\omega_0\tau_r)}{(0.5n\omega_0\tau) \cdot ...
0
votes
2answers
44 views

Why does $\sum^{n-1}_{j=i+1}j=\frac{(n+i)(n-i-1)}{2}$

Why does $\sum^{n-1}_{j=i+1}j=\frac{(n+i)(n-i-1)}{2}$? We know that $\sum^{n}_{j=1}j=\frac{(n+1)(n)}{2}$, but how does this standard one transform to the above result. Could someone help?
0
votes
1answer
29 views

How can I prove $||A||_1=$ max$_{1\le j \le n}$ $\sum_{i=1} ^n$ $|A_{ij}|$ ??

How can I prove $||A||_1=$ max$_{1\le j \le n}$ $\sum_{i=1} ^n$ $|A_{ij}|$ ?? $A \in R^{n\times n}$ , $x\in R^n$ The definition of 1 - norm of matrix is: $||A||_1$ = max$_{x\ne 0}$ ...
0
votes
2answers
44 views

Limit of Sum $\lim\limits_{n\to+\infty}n\sum\limits_{k=1}^{n}(n^2+k^2)^{-1}$ [duplicate]

Problem: $$\lim_{n\to +\infty} n\left(\sum_{k=1}^{n}\frac{1}{n^2+k^2}\right)=$$ Source: AP Calculus BC This looks like the definition of a definite integral to me, so I compared it to that equation. ...
0
votes
1answer
9 views

Name for inequality about sums of exponents with same base

Is there a name for the following inequality regarding sums of exponents which share a base? $$\text{For all integers $b \geq 2$, $n \geq 1$,} \\ \sum_{i=0}^{n-1}{b^i} < b^n$$
1
vote
0answers
29 views

Model linearly: Determine amount of units for production

A company produces 2 products in a week. Let $x_i$ denote the number of units of product $i$ to produce. Each product requires liters of Chemical X to make. Info is given below: \begin{array}{|c|c|} ...
0
votes
2answers
39 views

How to calculate $\sum^{n-1}_{i=0}(n-i)$?

How to calculate $\sum^{n-1}_{i=0}(n-i)$? $\sum^{n-1}_{i=0}(n-i)=n-\sum^{n-1}_{i=0}i=n-\sum^{n}_{i=1}(i-1)=2n-\frac{n(n+1)}{2}$ I am sure my steps are wrong. Could someone show me how to correct the ...
0
votes
1answer
59 views

Calculate the upper sum $U_n$ and lower sum $L_n$, on a regular partition of the following interval.

As the title says, I am trying to calculate the general formulas for the following integral. $$\int_0^1 (5+6x^2)dx$$ I've already completed a prior integral using the advice in this question, but I ...
1
vote
3answers
36 views

Can I approximate a series as an integral to find its limit and determine convergence?

Find $\lim \limits_{n \to \infty} (a_n)$, where $a_n=\frac{1}{n^2}+\frac{2}{n^2}+\frac{3}{n^2}+...+\frac{n}{n^2}$. So I can solve it like that ...
0
votes
1answer
92 views

Does this infinite sum converge?

Does this infinite sum converge? I have tried many methods that I know. $$\sum_{n=1}^\infty\frac{\left(1-\frac1n\right)^{n^2}}{3n^2+2}e^n$$ edit : after I saw the comments, I tried the ratio method ...