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

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3
votes
5answers
149 views

How do I succinctly note the sum of $(n-1)+(n-2)+…$?

I was playing with numbers and wanted to see how many possible connections there are in a network of $n$ nodes. I found that the answer was equal to ...
2
votes
1answer
49 views

Absolute Value Trig Sum

I have been trying to solve $$y(x)=\sum_{k=1}^{\infty} \frac{|\cos(kx)|}{k}$$ however, this is proving to be more difficult than I had hoped, and cannot seem to figure this out. What I have figured ...
0
votes
2answers
34 views

Prove convergence by considering the partial sums

Let $p$ be a non-zero natural number. Prove by considering the partial sums that $\sum \frac{1}{k(k+p)}$ converges. What is $\sum\limits_{k=1}^{\infty} \frac{1}{k(k+p)}$ No idea. Obviously, it ...
0
votes
1answer
26 views

Algebra question $n\sum_{k=0}^n {n-1 \choose k-1} p^{k}q^{n-k} = np\sum_{k=1}^n {n-1 \choose k-1} p^{k-1}q^{n-k}$

In a proof in my textbook one step goes from ... $$n\sum_{k=0}^n {n-1 \choose k-1} p^{k}q^{n-k} = np\sum_{k=1}^n {n-1 \choose k-1} p^{k-1}q^{n-k}$$ I understand that you can take the $p$ out because ...
10
votes
4answers
285 views

Which expansion of $e$ is more accurate?

We have two forms of $e^x$ $$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+....$$ and $$e^x=\frac{1}{\displaystyle 1-x+\frac{x^2}{2!}-\frac{x^3}{3!}+....}$$ The second form comes from ...
0
votes
2answers
77 views

If $\sum a_n$ converges, does $\sum a_n / 2^n$ converge as well?

If $\sum a_n$ converges, does $\sum \dfrac{a_n}{2^n}$ converge as well? I can't use differential or integral calculus for this.
2
votes
0answers
39 views

How the second form of following equation is derived form first form (i.e. given first line, what are the steps involved in writing second line

How the second form of following equation is derived form first form (i.e. what are the steps involved in writing second line)
1
vote
2answers
73 views

Is there any way to approximate a sum of square roots

I am trying to calculate a sum of square roots $\sum\limits_{i=1}^n \sqrt{a + i}$ and after some struggling and googling I gave up on this. Is there any way to get a closed formula for this sum ...
3
votes
2answers
50 views

Summation of trigonometric functions such as $\sin x$

I am currently studying Integration (a very basic introduction) and I have a question regarding the summation of trigonometric functions. Given $f(x) = \sin x$, determine the area under the curve ...
1
vote
5answers
65 views

The limit of a sum of the form $a_0 \sqrt n + a_1 \sqrt {n + 1} +\cdots + a_k \sqrt {n + k}$

If $ a_0 ,a_1 ,\ldots,a_k $ are real numbers such that $$ a_0 + a_1 + \cdots + a_k = 0$$ Find $$ \lim_{n \to \infty } (a_0 \sqrt n + a_1 \sqrt {n + 1} +\cdots + a_k \sqrt {n + k} )$$ I just ...
18
votes
1answer
657 views

How to prove a double sum is always an integer?

I have verified the following double sum is always an integer for $s$ up to $1000$ via Maple. But I can not prove it. Proofs, hints, or references are all welcome. Thanks! ...
1
vote
0answers
42 views

How should i go about proving an expression of this kind?

Lets say i have a complete bell polynomial that is equal to a summation such that $$ B_n(d_1,d_2,\cdots,d_n) = \sum_{k=0}^{n}[g(x)^{-k} h(k)] $$ Where $d_n = \frac{d^n}{dx^n}[f(x)\ln(g(x)]$ and ...
3
votes
1answer
48 views

Prove difference of summations $=\frac{e^2}{2}$

How do I prove that \begin{align} ...
1
vote
1answer
29 views

What is $s_3$ and $s_4$ for $x$

$\sum_{i=0}^n i^k = s_k(n)$, $s_k$ polynomial from degree $k+1$ I have already shown for $s_2(x) = \frac{x(x+1)(2x+1)}6$ How from the sum and $s_2(x)$ can be shown for $s_3(x)$ and $s_4(x)$ ...
0
votes
2answers
36 views

What is the value of the following summation?

Compute $$\displaystyle\sum \limits_{n=0}^\infty (-1)^{n+1} \frac{1}{9^n(2n+2)}$$ I am given the fact that $$ \frac{1}{2}\ln(1+x^2) = \sum \limits_{n=0}^\infty (-1)^n\frac{x^{2n+2}}{2n+2} $$ ...
1
vote
2answers
81 views

Using mathematical induction to prove $\frac{1}1+\frac{1}4+\frac{1}9+\cdots+\frac{1}{n^2}<\frac{4n}{2n+1}$

This induction problem is giving me a pretty hard time: $$\frac{1}1+\frac{1}4+\frac{1}9+\cdots+\frac{1}{n^2}<\frac{4n}{2n+1}$$ I am struggling because my math teacher explained us that in ...
1
vote
4answers
66 views

Compute $\lim\limits_{n \to \infty }\frac{981}{n+5}\sum_{i=1}^{n} \left (\frac{i^2}{n^2} \right)$

Compute the given limit $$ \lim_{n \to \infty }\frac{981}{n+5}\sum_{i=1}^{n} \left (\frac{i^2}{n^2} \right) $$ The sum is: Can someone please show me the steps to complete this problem? The answer ...
1
vote
1answer
72 views

Induction of closed form of summation

On Wikipedia the following closed form is derived - Generalised formula Can someone explain how the closed form below is derived? Edit Solution thanks to graydad
0
votes
2answers
34 views

Sum of a multi index series (really dumb question)

$$\sum_{\substack{i,j=1 \\ i \neq j}}^{l} (x_iy_i + x_jy_j) = k \sum_{i=1}^l x_iy_i$$ I have to find $k$. I know the question is really stupid, but for some reason I am unable to solve this.
2
votes
0answers
58 views

bessels equation

A long cylinder (radius r =b) initially at T=f(r) is exposed to a cooling medium which extracts heat uniformly from its surface. It was assumed that heat transfer takes place only through radial ...
4
votes
2answers
62 views

Show that the series $\sum \frac{\sin \left(\frac{\left( 3-4n \right)\pi }{6}\right) }{2^{n}}$ converges?

Using the addition formula for the sine function I have managed to reduce this to a simpler form: $$\sum \frac{\cos \frac{2n\pi }{3}}{2^{n}}$$ It is obvious here that it passes the n-th term ...
2
votes
2answers
56 views

Can you help me simplify this summation notation?

$$\sum_{i=1}^n \frac{n}{n+1}i^2$$ and $$\sum_{i=1}^n \frac{i}{n}$$ (n is a constant)
1
vote
1answer
44 views

Proving Cauchy-Schwarz related proof using induction

So the first thing I was asked to prove was this: If $a_1,a_2,...,a_n$ and $b_a,b_2,...,b_n$ are real numbers, use induction to show. ...
0
votes
1answer
16 views

Inequality involving different diameter average

I have found an assertion in a scientific book (Hinds, Aerosol Technology, 2nd Edition, 1998, p. 83-84) that claims: Given the general form [here for grouped data] for the diameter of an average ...
7
votes
2answers
283 views

Finding an inverse trigonometric sum

How do I prove that the following equality holds- $$\sum_{p=1}^{10} \sum_{q=1}^{10} \arctan \left(\dfrac{p}{q}\right)=25\pi$$ I tried to create telescoping terms by using the $\arctan{A}-\arctan{B}$ ...
1
vote
3answers
43 views

Evaluate $\sum_{k=400}^{2000} \frac {2^{3-4k}} {8^{2k+3}}$

Evaluate $$\sum_{k=400}^{2000} \frac {2^{3-4k}} {8^{2k+3}}$$ So far, I was able to get to $$\frac{1}{64}\sum_{k=400}^{2000} \frac {1} {8^{2k}\cdot2^{4k}}$$ And then I'm completely stuck.
0
votes
2answers
21 views

comparison test to show that $\sum_{n=1}^{\infty}\frac{1}{(n+2)\sqrt{ \ln ^3(n+3)}}$ converges

As the title says, I know that this sum converges and I want to find a suitable comparison test. Cauchy's root test and d'Alembert's ratio test gave inconclusive results. According to wolfram this ...
-1
votes
2answers
42 views

Is the series $\sum \frac{3 + \sin n}{n^2}$ convergent?

How can I show if the following series converges? $$\sum \frac{3 + \sin n}{n^2}$$ I can't use differential or integral calculus (hasn't been covered in my class yet.)
1
vote
4answers
85 views

Sum of all triangle numbers

Does anyone know the sum of all triangle numbers? I.e 1+3+6+10+15+21... I've tried everything, but it might help you if I tell you one useful discovery I've made: I know that the sum of ...
0
votes
3answers
53 views

How to do this limit?

$$\large\lim_{n\to \infty}\large\frac{\sum_{k=1}^n k^p}{n^{p+1}}$$ I'm stuck here because the sum is like this: $1^p+2^p+3^p+4^p+\cdots+n^p$. Any ideas?
2
votes
4answers
123 views

Is $\sum\frac{1}{\sqrt{n+1}}$ convergent or divergent?

$$\sum\frac{(-1)^n}{\sqrt{n+1}} \text{and} \sum\frac{1}{\sqrt{n+1}}$$ The first one is an alternating series, so it would just be: $$\sum (-1)^n\frac{1}{\sqrt{n+1}}\Rightarrow ...
8
votes
2answers
90 views

Reworking $\sum_{n \leq x} \frac{1}{n^s}$, where $n$ is relatively prime to some fixed $k$

For a fixed integer $k \geq 1$ and real $s>0$ I want to rework the partial sums $$\sum_{\substack{ n \leq x \\ \text{gcd}(k,n) = 1 }} \frac{1}{n^s}$$ in such a way that I can find an ...
2
votes
1answer
31 views

Notation of sigmas

When one writes $$\sum_{i+j=4} a_i a_j$$ is that then equal to $$a_0a_4 + a_1a_3 + a_2^2$$ or $$\sum_{i=0}^4 a_i a_{4-i}=2a_0a_4+2a_1a_3 + a_2^2$$ I suddenly got confused while writing these sigmas ...
0
votes
1answer
33 views

Notation: $\sum\limits_{i=m}^n i$, with $m>n$.

I came across with that notation in one book. They defined $$\overline{X_n}=\frac{X_1+...+X_n}{n}.$$ Then they define $n\in \mathbb N$, $k_n\in\mathbb Z$ such that: $$[k_n]^2\leq n\leq ...
3
votes
0answers
41 views

Evaluating $\sum\limits_{k=0}^n\cos(kx)$ and $\sum\limits_{k=0}^n\sin(kx)$ without Complex Numbers [duplicate]

Alright, so the standard way to evaluate $\sum\limits_{k=0}^n\cos(kx)$ and $\sum\limits_{k=0}^n\sin(kx)$, is to respectively take the real and imaginary part of $$\sum_{k=0}^n{\rm e}^{ikx}={\frac ...
3
votes
0answers
72 views

If $A = \frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\cdots+\frac{1}{\sqrt{999}}+\frac{1}{\sqrt{1000}}.$ Then $\lfloor A \rfloor$ is,

If $\displaystyle A = \frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\cdots\cdots\cdots+\frac{1}{\sqrt{999}}+\frac{1}{\sqrt{1000}}.$ Then $\lfloor A \rfloor$ is, where $\lfloor A\rfloor = A-\{A\}.$ ...
1
vote
2answers
53 views

Difference between equals/approaches/approximate

Consider the series $$\sum\limits_{k=0}^{\infty} \frac{1}{2^k} = 2$$ Is it correct to say "$\text{the series approaches 2 ?}$" if so, shouldn't we replace $=$ with $\approx$ ? Also Is it ...
1
vote
0answers
37 views

To find the approximate solution of the series for large N

to make sum of series including combinations ${N\choose 1}{N\choose 0}+{N \choose 2}{N\choose 1}a^2 b^{-2} + {N\choose 3}{N \choose 2}a^4 b^{-4}+{N\choose 4}{N \choose 3}a^6 b^{-6}+...$Is it possible ...
3
votes
3answers
54 views

Evaluating $\sum_{n = 1}^{\infty} \frac{2}{2^{n}}$

Evaluate $$\sum_{n = 1}^{\infty} \frac{2}{2^{n}}$$ This is a geometric series and since $a = \dfrac{1}{2}$ Then the infinite sum is jsut $S = \dfrac{1}{1-\frac{1}{2}} = 2$ Then I multiply by $2$ ...
2
votes
1answer
38 views

Trigonmetric sum of inverses

Prove that: $$\sum^{45}_{k=1}\frac{1}{\cos1^\circ-\cos(87+4k)^\circ}=\frac{1}{2\sin 1^\circ}$$ Numerically, this is accurate comparing the lhs and rhs. Some ideas: We can transform the question ...
1
vote
1answer
80 views

Prove that $(1+2+3+\cdots+n)^2=1^3+2^3+3^3+\cdots+n^3$ $\forall n \in \mathbb{N}$. [duplicate]

Prove that $(1+2+3+\cdots+n)^2=1^3+2^3+3^3+\cdots+n^3$ for every $n \in \mathbb{N}$. I'm trying to use induction on this one, but I'm not sure how to. The base case is clearly true. But when I add ...
4
votes
2answers
45 views

Evaluating $\sum_{k=0}^{\infty}\frac{1-2^k}{3^k}$

I am trying to find the sum of following infinite series $$\sum_{k=0}^{\infty}\frac{1-2^k}{3^k}$$ I tried starting the problem with rewriting it as $\dfrac13+\left(\frac{-2}{3}\right)^k$, am i ...
2
votes
3answers
76 views

Show that $\sum_{n=0}^{\infty} \dfrac{n}{2^{n+1}} = 1$

My Work I felt the best way to go about this problem was to compare it to a well known MacLaurin series. I noticed it resembled the reciprocal of the absolute value of the MacLaurin series of ...
0
votes
2answers
53 views

Sum function operation: coefficient.

I have problem with the sum: $$ \sum_{k=0}^n \dbinom{n}{k}(\cos \alpha)^k(i\sin \alpha)^{n-k}\,\, $$ Apparantly, I have an imaginary unit therefore I need to distinguish even and odd powers of $i$ to ...
1
vote
1answer
108 views

Strict upper and lower bounds of a sum (Big-Theta)

I am trying to find a function f(k) such that $S_k=\sum_{n=1}^{k^2-1}(\lfloor\sqrt{n}\rfloor)=\Theta(f(k))$. What I have done so far: Ignoring the floor asymptotically we get: ...
0
votes
0answers
47 views

Sum to infinity of the sum 1/n^2 [duplicate]

In my textbook it's mentioned that the sum $\lim\limits_{n\rightarrow\infty}(\sum\limits_{i=1}^n1/i^2=\pi^2/6)$. But how would you arrive at this result?
0
votes
1answer
26 views

Show $1+2^2+…+n^n$ ≤ $(1+1/n-1)*n^n$

$1+2^2+...+n^n$ ≤ $(1+1/(n-1))*n^n$ Well what I come up with is it's left to prove $1+2^2+...+n^{n-1}$≤ $n^n/(n-1)$ I think I need to somehow come up with a summation of the former then compare it ...
2
votes
1answer
166 views

Interpretation of Ramanujan summation of infinite divergent series

I am not mathematician by any means so this question might be rather stupid. I came across this Wikipedia article on Ramanujan's summation and found this bewildering formula, 1 + 2 + 3 + ... = - ...
2
votes
1answer
32 views

How does this pattern work?

I know that $$ \sum_{k=0}^{\infty}\frac{1}{k\,!}=e=\lim_{n \to \infty}{(1+\frac{1}{n})^n} $$ but why $$ \sum_{k=0}^{\infty}\frac{1}{(2k)\,!} = \cosh(1) $$ and $$ \sum_{k=1}^{\infty}\frac{1}{(2k+1)\,!} ...
0
votes
1answer
40 views

Further explanation regarding calculation of E[X^2]

I was reading over the following evaluation of $ E[X^2] $ on the following pdf: http://crab.rutgers.edu/~guyk/dmlec/lectures/lec15/l15.pdf. This part was especially confusing for me: ...