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

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0answers
13 views

Solving an Integral by Summation

My final answer for this question was 90 but I'm not quite sure if I'm even doing it right... I was wondering if anyone could help me solve this for me to check my work against. ...
0
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2answers
56 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
48 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
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1answer
32 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
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2answers
30 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.
0
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0answers
26 views

bessels equation [on hold]

How to solve this equation? $$\sum_{m=1}^{\infty} \frac{1}{\beta_m}\frac{\mathrm{J_1}(\beta_m\cdot b)\mathrm{J_0}(\beta_m\cdot r))}{\mathrm{J}_0^2(\beta_m\cdot b)+\mathrm{J}_1^2 (\beta_m\cdot b)} ...
1
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2answers
41 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)
0
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1answer
30 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
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1answer
15 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
253 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
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3answers
41 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
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2answers
18 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 ...
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2answers
35 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.)
0
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1answer
34 views

Given that the sum of a series $a_n$ converges, does the following sum converge? [on hold]

Given a series sum $\sum a_n = a_1 + a_2 + \dotsc$ converges and has a total sum of $s$. Does the following sum converge? $$\sum b_n \text{ where } b_n = a_{3n-2} + a_{3n-1} + a_{3n}$$ If it does ...
1
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4answers
58 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
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3answers
49 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
104 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
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2answers
78 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 ...
1
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0answers
9 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 ...
2
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4answers
51 views

Determining whether $\sum_{k=1}^\infty \frac{x^k}k$ converges [on hold]

$$\sum_{k=1}^\infty \frac{x^k}k$$ Does this series converge, if yes, then for what values of $x$?
0
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1answer
28 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
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0answers
23 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
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0answers
62 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
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2answers
34 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
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0answers
31 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
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3answers
50 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
30 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
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1answer
55 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
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2answers
41 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
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3answers
73 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
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2answers
37 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 ...
0
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1answer
24 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
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0answers
46 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
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1answer
24 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 ...
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0answers
21 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
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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
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1answer
38 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: ...
1
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1answer
26 views

Evaluation of $\sum\limits_{i = 0}^s {\left( {\begin{array}{*{20}{c}} q \\ {{2^i}} \end{array}} \right)}$

Assume that $q=2^s$ for some non-negative integer $s$. Is there any simple formula for: $$\sum\limits_{i = 0}^s {\left( {\begin{array}{*{20}{c}} q \\ {{2^i}} \end{array}} \right)}?$$
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0answers
29 views

The coefficient of $t^n$ in $\left(\sum_{k=1}^{n-1} t^k\right)^r$

I'm trying to count the number of ways of writing a general natural number $n\geq 2$ as the sum of $r$ smaller numbers where each of these numbers is at least $2$ - that is, I want to count the number ...
1
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4answers
94 views

How do I express the sum $(1+k)+(1+k)^2+(1+k)^3…+(1+k)^N$ for $|k|<<1$ as a series?

Wolfram Alpha provides the following exact solution $$ \sum_1^N (1+k)^i = \frac{(1+k)\,((1+k)^N-1)}{k}.$$ I wish to solve for $N$ of the order of several thousand and $|k|$ very small (c. $10^{-12}).$ ...
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0answers
29 views

Sigma Notation for Power

How do I evaluate the telescopic sum for these two? (You do not have to provide the answer if you feel like you dont want to) I never worked with sigma notation regarding powers. Question 1 ...
5
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2answers
221 views

Alternating Series , why start at n = 1?

$$\sum_{n=1}^\infty(-1)^nb_n$$ Convergent if $b_{n+1} \le b_n$ and if $\lim b_n = 0$ I'm learning taylor series now , and I'm confused with this alternating series test , I've searched around and ...
0
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1answer
13 views

Reducing summation in recurrence relation

I am trying to solve this recurrence relation: $T(n) = 7T(\frac{n}{2}) + 18(\frac{n}{2})^2$ which is for Strassen's fast matrix multiplication. However I am stuck on trying to reduce the summation. I ...
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0answers
35 views

Math basic problem [closed]

OK I explain the problem: I have one printer. The driver(software) installed in my computer can print in different modes: ...
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0answers
48 views

close form solutions for infinite sums

I am interested in finding the following infinite sums in order to reparameterize a distribution function. $\sum_{n=0}^{\infty} \theta^{n}q^{n(n-2)/2} $ where $\theta>0$ and $1>q>0$ and ...
1
vote
1answer
16 views

Amount of Pharmaceutical Left in the Patient's Body After the nth Dose is Administered

A patient is administered 500 mg of a certain pharmaceutical every 6 hours. The half-life of the pharmaceutical in the body is 130 minutes. Determine the amount P(n) of the pharmaceutical that remains ...
5
votes
3answers
35 views

Find the summation $\sum_{k=0}^n (-1)^k \binom{n}{k}\frac{1}{s+k}$

Anyone can help me finding this summation: $$ \sum_{k=0}^n (-1)^k \binom{n}{k}\frac{1}{s+k}. $$ Where there is a similar one with known answer $$ \sum_{k=0}^n (-1)^k ...
0
votes
2answers
18 views

How to mathematically describe the number of Element x in a set

I am trying to formulate the following. I have a Set A={x, y, z}, I also have a Set B, C and D, which all are subsets of A. It is not exactly defined which elements are in B, C and D. I only want to ...
0
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0answers
6 views

Sum of values of a function with concurrent lines

Lets say I have a linear function as follows: $y=k*x_0$, where $k$ is NOT a constant. Values of $k$ can be between $0$ and $k_1$. In this example values of $k$ range from $k_2$ to $k_3$. My ...
0
votes
2answers
27 views

Proving Primness in a summation

I've been hitting my head against the wall for a little bit trying to figure out where to get started on proving (or disproving) the following: $\exists k \in \mathbb{Z} $ such that$ ...