Sum Definition and 1000 Threads

I'm using the sum of a geometric series formula, but I'm not sure how to find the ratio, r. The n is confusing me. The solution is below, but I'm having trouble with the penultimate step.

Couldn’t arrive to the answer

Cannot conclude the answer

Hi, I'm trying to solve the sum of following infinite series: \sum_{k=1}^{\infty} \frac{{k}^{2}+4}{{2}^{k}} = \sum_{k=1}^{\infty} \frac{{k}^{2}}{{2}^{k}} + \sum_{k=1}^{\infty} \frac{4}{{2}^{k}} Using partial sum we can rewrite the first series: \sum_{k=1}^{\infty}...

I tried by ##S=1+(1/1!)(1/4)+(1.3/2!)(1/4)^2+...## ##S/4=1/4+(1/1!)(1/4)^2+(1.3/2!)(1/4)^3..## And then subtracting the two equations but i arrived at nothing What shall i do further?

My question is Why is the sum to infinity used as opposed to Sum to n? and How can I deduce that the sum to infinity must be used from the question?Total Distance = h + 2*Sum of Geometric progression (to infinity) h + 2*h/3 / 1-1/3 h + 2h/3 *3/2 = h + h = 2h At first I did sum to infinity...

https://www.quantamagazine.org/sum-of-three-cubes-problem-solved-for-stubborn-number-33-20190326/ has the details

If one shows that ##U\cap V=\{\textbf{0}\}##, which is easily shown, would that also imply ##\mathbf{R}^3=U \bigoplus V##? Or does one need to show that ##\mathbf{R}^3=U+V##? If yes, how? By defining say ##x_1'=x_1+t,x_2'=x_2+t,x_3'=x_3+2t## and hence any ##\textbf{x}=(x_1',x_2',x_3') \in...

v_1 = <-8/21,-20/21> v_2 = <50/21,-20/21> When I take the dot product of v_2 and <2,5> I get zero, indicating they are perpendicular. Sorry for the hand writing.

Show that \[\tan^{-1}(k) = \sum_{n=0}^{k-1}\tan^{-1} \left ( \frac{1}{n^2+n+1} \right ),\;\;\;\;\; k \geq 1,\] - and deduce that \[ \sum_{n=0}^{\infty}\tan^{-1} \left ( \frac{1}{n^2+n+1} \right ) = \frac{\pi}{2}.\]

I am reading an interesting book by Julian Havil called:" Gamma-Exploring Euler's Constant." Much of the book is devoted to the harmonic series,a slowly diverging series that tends toward infinity.However,one paragraph puzzles me. On p. 23 he says: " In 1968 John W. Wrench Jr calculated the...

I don't know how to show that this limit is zero. It seems that ##\sum_{i=1}^N a_{i,N} /N = 1## and the fact that ## 0 < a_{i,N} < M > 1## implies that some ##a_{i,N}## are less than one. Another conclusion I guess is correct to draw is that ##\lim_{N \to \infty} \sum_{i=1}^N a_{i,N}^2 /N < 1##.

Let $P_i$ denote the $i$thpoint on the surface of an ellipsoid: $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2} = 1$, where the principal semiaxes obey: $0 < a < b < c$. Maximize the sum of squared distances: \[\sum_{1\leq i < j \leq 2n}\left | P_i-P_j \right |^2\] - over alle possible...

Hi, Although I'm using trigonometric form of Fourier transform, first I'd discuss both, exponential and trigonometric forms, for the sake of context. Now proceeding toward the main question and we would only be using trigonometric form. % file name...

For part a, A force must be applied so that the entire mass can be 'held up'. Therefore the necessary force must be equal to the gravitation force on all the objects: m(rod) * 9.8 + m(LB) * 9.8 + m(RB) * 9.8 = 137 N For part b, (This is where I'm confused) let's set the point at which the...

Two KL functions $f_1:\mathbb{R}^n\rightarrow \mathbb{R}$ and $f_2:\mathbb{R}^n\rightarrow \mathbb{R}$ are given which have KL exponent $\alpha_1$ and $\alpha_2$. What is the KL exponent of $f_1+f_2$?

For instance, here is an example from my own simulations where all underlying signals follow the same analytical law, but they have random phases and amplitudes (such that the sum of the set is 1). The thick line represents the sum: Clearly, the sum tends to progressively get flatter as ##N...

In "Sheldon Axler's Linear Algebra Done Right, 3rd edition", on page 21 "internal direct sum", or direct sum as the author uses, is defined as such: Following that there is a statement, titled "Condition for a direct sum" on page 23, that specifies the condition for a sum of subspaces to be...

I understand that the energy of an electric field arises from the work put into gathering the electrons together to create the field. Bringing electrons close together requires energy because they naturally want to repel. This potential energy is stored in the field itself and the field has an...

I can construct examples that are less than or equal to ##4n## quite easily, but for the life of me I cannot come with example where it's greater than

Hello, I want to plot this PDE which is non homogeneous: ut=kuxx+cut=kuxx+c u(x,0)=c0(1−cosπx)u(x,0)=c0(1−cosπx) u(0,t)=0u(1,t)=2c0u(0,t)=0u(1,t)=2c0 I have a code that can solve this problem and plot it with those boundary and initial conditions but not with the non homogeneous term...

I found the first 4 terms of the series: ½-(1/16)x^2+(1/64)x^4-(7/1536)x^6. I cannot however simplify this to a sum. the 7 in the numerator of the last term of the above expansion is the sticking point.

A0 = 1 A1 = 3 3(An-1) / 4(An-2) = An

Hello, I am working through Spivak for self study and sharpening my math skills. I have become stuck on an exercise. What I need to show is the following: $$ (a + b) \sum_{j = 0}^{n} \binom nj a^{n-j}b^{j} = \sum_{j = 0}^{n + 1} \binom{n+1}{j} a^{n-j + 1}b^{j} $$ My attempt, starting from...

Hi, I am self studying induction and came across the following problem. I am stuck on how to proceed (I need to use induction, I know there is a direct proof). My proof attempt is as follows: Let ## P (m) ## be the proposition that: $$ \sum_{i = m + 1}^{n} i = \frac{(n - m)(n + m + 1)}{2} $$...

Let $a_1,a_2, ... , a_n$ be positive numbers. Let $i_1,i_2, ... , i_n$ be a permutation of $1,2,...,n$. Determine the smallest possible value of the sum: $$\sum_{k=1}^{n}\frac{a_k}{a_{i_k}}$$

Curious about proving that ##\sum_{m=2}^\infty \sum_{n=2}^\infty 1/n^m ## = 1 ran this in Matlab and n,m to 2:1000 =0.9990, and n,m 2:10000 =0.9999, so it does appear to converge to 1

The difference between light and very heavy atoms reflects itself in these two schemes. My question is why one scheme for the vector sum is necessarily the right & suitable sum model for one case, and the 2nd scheme suits the 2nd case ? In other words, why & how the relative magnitude of the...

Prove the existence of a positive integer divisible by $2019$ the sum of whose decimal digits is $2019$.Source: Nordic Math. Contest

Homework Statement I have encountered this problem from the book The Physics of Waves and in the end of chapter six, it asks me to prove the following identity as part of the operation to prove that as the limit of ##W## tends to infinity, the series becomes an integral. The series involved is...

Hello there, I'm working on a kinetic theory of mixing between two species - b and w. Now, if I want to calculate the number of different species B bs and W ws can form, I can use a simple combination: (W+B)!/(W!B!) Now, in reality in my system, ws and bs form dimers - ww, bb, wb and bw...

##\sum_{n=1}^\infty 1/n^2 ## converges to ##π^2/6## and every other series with n to a power greater than 1 for n∈ℕ convergesis it known if the sum of all these series - ##\sum_{m=2}^\infty \sum_{n=1}^\infty 1/n^m ## for n∈ℕ converges? apologies for any notational flaws

Evaluate the following double sum of a product: $$\sum_{j=1}^{\infty}\sum_{n=1}^{\infty}\left(n\prod_{i=0}^{n}\frac{1}{j+i}\right)$$

Greg Bernhardt submitted a new blog post The Sum of Geometric Series from Probability Theory Continue reading the Original Blog Post.

Let us have some localized density of sources, S, in a plane, each of which produces a localized circular vector field. Let us work in polar coordinates. Let the density of sources, S = Aexp(-r^2/a^2) and let each source have circular vector field whose strength is given by exp(-(r-r_i)^2/b^2)...

Homework Statement [/B] From the Rodrigues’ formulae, I want to derive nature of the spherical Bessel and Neumann functions at small values of p. Homework Equations [/B] I'm going to post an image of the Bessel function where we're using a Taylor expansion, which I'm happy with and is as far...

Homework Statement Given: Ψ and Φ are orthonormal find (Ψ + Φ)^2 Homework Equations None The Attempt at a Solution Since they are orthonormal functions then can i do this? (Ψ + Φ) = (Ψ + Φ)(Ψ* + Φ*)?

Hi! $$(x_{n})_{n\geq 2}\ \ x_{n}=\sqrt[n]{1+\sum_{k=2}^{n}(k-1)(k-1)!}$$ $$\lim_{n\rightarrow \infty }\frac{x_{n}}{n}=?$$ I know how to solve the limit but I don't know how to solve the sum $\sum_{k=2}^{n}(k-1)(k-1)!$ which should be $(n! - 1)$ The limit would become $\lim_{n\rightarrow \infty...

PLEASE HELP1.What is the sum of the measures of the interior angles of a heptagon? A. 1260∘ B. 2520∘ C. 900∘ D. 1800∘ my answer is C 5.If the sum of the interior angle measures of a polygon is 3600∘, how many sides does the polygon have? A. 22 sides B. 20 sides C. 18 sides D. 10 sides MY...

For d binary digits, the number of ways W to get sum s is: W(d,s) = d! / (s! * (d-s)!) Are there similar formula(s) for n-ary digits?

Homework Statement Express the sum as a fraction of whole numbers in lowest terms: ##\frac{1}{1⋅2}+\frac{1}{2⋅3}+\frac{1}{3⋅4}+...+\frac{1}{n(n+1)}## Homework EquationsThe Attempt at a Solution Please see attached image for my work. The reason I am posting the image rather than typing this...

How would you, personally, do this summation the quickest way?

Homework Statement [/B] Homework Equations Drawing a diagram for the forces is the easy part. I am not sure I am doing the equation of the sum of the torques well. The Attempt at a Solution This is my attempt for the forces[/B] And this for the torques:

Evaluation of $\displaystyle \sum^{n}_{k=0}\binom{n+k}{k}\cdot \frac{1}{2^k}$

<Moderator's note: Moved from a technical forum and thus no template.> Let there be two vectors, u and v. Whose magnitudes are constant u = [a, b] v = [x, y] Define c = ||u|| and k = ||v|| Now sum the vectors: w = u + v = [a, b] +[x, y] = [a+x, b+y] Now find ||w|| ||w|| =√(a+x)2+(b+y)2...

Calculation of probability with arithmetic mean of random variables There are 4 people, each of whom has one deck of cards with 500 cards that are numbered from 1 to 500 with no duplicates. Each person draws a card from his deck and I would like to calculate the probability of the event that...

Hey! :o Let $V$ be a vector space. Let $b_1, \ldots , b_n\in V$ and let $\displaystyle{b_k':=\sum_{i=1}^kb_i}$ for $k=1, \ldots , n$. I want to show that $\{b_1, \ldots , b_n\}$ is a basis of $V$ iff $\{b_1', \ldots , b_n'\}$ is a basis of $V$. I have done the following: Let $B:=\{b_1...

Homework Statement "Given that the joint distribution of ##X## and ##Y## is ##f(x,y)=\frac{1}{2}(x+y)e^{-(x+y)},\text { for } x,y>0## and ##0## otherwise, find the distribution of ##Z=X+Y##." Homework Equations ##f_Z(z)=\int_{\mathbb{R}}f(x,z-x)dx##...

Homework Statement Find the sum of the series Homework EquationsThe Attempt at a Solution Not sure exactly where to start. If I move 3 outside the sum I'm left with 3*sigma(1/n*4^n), which I can rewrite to 3*sigma((1/n)*(1/4)^n), which party looks like a geometric series..Any tips?

Homework Statement Hello, I need to find an expression for the sum of the given power series The Attempt at a Solution I think that one has to use a known Maclaurin series, for example the series of e^x. I know that I can rewrite , which makes the expression even more similar to the...