Sum Definition and 1000 Threads

  1. anemone

    MHB Prove a sum is not the fifth power of any integer

    Suppose $X$ is a number of the form $\displaystyle X=\sum_{k=1}^{60} \epsilon_k \cdot k^{k^k}$, where each $\epsilon_k$ is either 1 or -1. Prove that $X$ is not the fifth power of any integer.
  2. I

    MHB Magnitude and angle of vector sum

    help! find the magnitude of the resultant force and the angle it makes with the positive x-axis. i don't have any examples in my book like this one
  3. Math Amateur

    MHB Direct Sum of n Vector Spaces Over F - Knapp Proposition 2.31 - Pages 61-62

    I am reading Chapter 2: Vector Spaces over $$\mathbb{Q}, \mathbb{R} \text{ and } \mathbb{C}$$ of Anthony W. Knapp's book, Basic Algebra. I need some help with some issues regarding Theorem 2.31 (regarding the direct sum of n vector spaces) on pages 61-62. Theorem 2.31 and its accompanying...
  4. Math Amateur

    MHB Universal Mapping Property of a Direct Sum - Knapp Pages 60-61

    I am reading Chapter 2: Vector Spaces over $$\mathbb{Q}, \mathbb{R} \text{ and } \mathbb{C}$$ of Anthony W. Knapp's book, Basic Algebra. I need some help with some issues regarding the Universal Mapping Property of direct sums of vector spaces as dealt with by Knapp of pages 60-61. I am not...
  5. I

    MHB Is This the Sum of the Series? $\sum_{n=0}^{\infty}\frac{2^n}{3^nn!}$

    $\sum_{n=0}^{\infty}\frac{2^n}{3^nn!}$ is this correct? $\sum_{n=0}^{\infty}(\frac{2}{3})^n \frac{1}{n!}$ $\sum_{n=0}^{\infty}\frac{(x)^n}{n!}=e^x$ $x=2/3$ $e^x=e^{2/3}
  6. N

    How to rewrite the provided sum in another form?

    Homework Statement How to get from Sum of 2(cos((3pi)/(2^(k+1)))sin(pi/(2^(k+1)))) from k = 1 to infinity to Sum of sin((4pi)/(2^(k+1))) - sin((2pi)/(2^(k+1))) from k = 1 to infinity The two expressions are equivalent. I need help getting from the first expression to the second.
  7. anemone

    MHB Prove a sum is a composite number

    For positive integers $p,\,q,\,r,\,s$ such that $ps=q^2+qr+r^2$, prove that $p^2+q^2+r^2+s^2$ is a composite number.
  8. T

    How Is the Sum Rule for Limits Proven?

    Prove the Sum Rule for Limits $$\lim_{x\to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x) = L + M$$ Proof Assume the following: $$\lim_{x \to a} f(x) = L, \space\lim_{x \to a} g(x) = M$$ Then, by definition ##\forall \epsilon_1 > 0, \exists \delta_1 > 0## such that...
  9. I

    MHB Find Sum of Series to Within 0.01

    how many term of the series $\sum_{n=2}^{\infty}\frac{1}{[n(ln (n))^2]}$ would you need to add to find its sum to within 0.01? approximate the sum of the series correct to four decimal places. $\sum_{n=1}^{\infty}\frac{(-1)^n}{3^nn!}$
  10. Y

    Determine the sum of the given series:

    Homework Statement Sum starting from n=1 to infinity for the expression, (3/4^(n-2)) What the solutions manual has done is multiply the numerator and the denominator by 4. 12/(4^(n-1)) I don't know what they have done from here on: 12 / (1 - 1/4) = 16 Can someone...
  11. A

    MHB Convergence of a Series: Is My Approximation Accurate Enough?

    Hey guys, I just wanted to run a quick series question by you guys just to confirm my answer. I'm doubting whether or not I should keep going or if S6 is enough. I got S5 = -0.28347 and S6 = -0.28347, so that is where I concluded than Sn ~ -0.2835. I would appreciate it if someone could...
  12. B

    MHB Sum Binomials: Proving Numerical Test Result

    I have this sum $$\left(N+1\right)^{2}\underset{j=1}{\overset{N}{\sum}}\frac{\left(-1\right)^{j}}{2j+1}\dbinom{N}{j}\dbinom{N+j}{j-1}\underset{i=1}{\overset{N}{\sum}}\frac{\left(-1\right)^{i}}{\left(2i+1\right)\left(i+j\right)}\dbinom{N}{i}\dbinom{N+i}{i-1}$$ and numerical test indicates that is...
  13. T

    Direct Sum and Direct Product: Understanding the Differences in Vector Spaces

    The definition (taken from Robert Gilmore's: Lie groups, Lie algebras, and some of their applications): We have two vector spaces V_1 and V_2 with bases \{e_i\} and \{f_i\}. A basis for the direct product space V_1\otimes V_2 can be taken as \{e_i\otimes f_j\}. So an element w of this space...
  14. anemone

    MHB Find the sum of all real solutions

    Find the sum of all real solutions for $x$ to the equation $\large (x^2+4x+6)^{{(x^2+4x+6)}^{(x^2+4x+6)}}=2014$. P.S. I know this doesn't count as a challenge(no matter how you slice it) because it's quite obvious and rather a very straightforward sort of problem but I'd like to share it...
  15. anemone

    MHB Find Sum of Diagonals of Pentagon $PQRST$

    Let $PQRST$ be a pentagon inscribed in a circle such that $PQ=RS=3$, $QR=ST=10$, and $PT=14$. The sum of the lengths of all diagonals of $PQRST$ equals to $\dfrac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Find $a+b$.
  16. J

    Proving the Induction Step: \sum_{j=1}^{k+2} j \cdot 2^j = (k+2)\cdot 2^{k+4}+2

    Homework Statement prove by induction \sum_{j=1}^{n+1} j \cdot 2^j = n \cdot 2^{n+2}+2; n \ge 02. The attempt at a solution P(0) \sum_{j=1}^{0+1} j \cdot 2^j = 0 \cdot 2^{0+2}+2 2+2 here is where I need some help is P(k) \sum_{j=1}^{k+1} j \cdot 2^j = (k+1) \cdot 2^{k+3}+2 ?? then...
  17. anemone

    MHB Sum Infinity Express: Rational Number Solution

    Express $\displaystyle \sum_{n=1}^{\infty}\sum_{m=1}^{\infty} \dfrac{1}{m^2n+mn^2+2mn}$ as a rational number.
  18. J

    Finding the sum of 1^3 + 2^3 + + n^3 by induction

    1^3+2^3+...+n^3 = \left[ \frac{n(n+1)}{2}\right]^2; n\ge 1 P(1) = 1^3 = \frac{8}{8} = 1 P(k) = 1^3+...+k^3 = \left[ \frac{k(k+1)}{2}\right]^2 (induction hypothesis) P(k+1) = 1^3+...+k^3+(k+1)^3 = \left[\frac{(k+1)(k+2)}{2}\right]^2 I start getting stuck here I foiled it out then let m =...
  19. J

    Write F as a sum of an orthogonal and parallel vector

    an object is moving in the direction i + j is being acted upon by the force vector 2i + j, express this force as the sum of a force in the direction of motion and a force perpendicular to the direction of motion. the parallel would be \hat{i}+\hat{j} and the orthogonal would be \hat{i} -...
  20. &

    Subdividing an integral into a sum of integrals over a given interval

    How does one prove the following: \int^{c}_{a} f\left(x\right)dx = \int^{b}_{a} f\left(x\right)dx +\int^{c}_{b} f\left(x\right)dx where f\left(x\right) is continuous in the interval x\in \left[a, b\right], and differentiable on x\in \left(a, b\right). My approach was the following...
  21. kaliprasad

    MHB Find Number b/w 1000-2000: Impossible Sum of Consec. Nums

    Find the number between 1000 and 2000 that cannot be expressed as sum of (that is >1) consecutive numbers.( To give example of sum of consecutive numbers 101 = 50 + 51 162 = 53 + 54 + 55 ) and show that it cannot be done
  22. Saitama

    MHB Evaluating a Sum Problem: Find Value

    Problem: Let $[x]$ be the nearest integer to $x$. (For $x=n+\frac{1}{2}, n\in \mathbb{N}$, let $[x]=n$). Find the value of $$\sum_{m=1}^{\infty} \frac{1}{[\sqrt{m}]^3}$$ Attempt: I tried writing down a few terms and saw that $1$ repeats $2$ times, $2$ repeats $4$ times but I didn't check it...
  23. kq6up

    Finding the Sum of an Infinite Series

    Homework Statement Find the expectation value of the Energy the Old Fashioned way from example 2.2. Homework Equations ##\left< E \right> =\frac { 480\hbar ^{ 2 } }{ \pi ^{ 4 }ma^{ 2 } } \sum _{ odds }^{ \infty }{ \frac { 1 }{ { n }^{ 4 } } } ## The Attempt at a Solution Never...
  24. anemone

    MHB What Is the Maximum Value of $pq+qr+rs$ When $p+q+r+s=63$?

    If $p,\,q,\,r,\,s$ are positive integers with sum 63, what is the maximum value of $pq+qr+rs$?
  25. S

    Finding sum of infinite series

    Homework Statement Recognize the series $$3-3^3/3!+3^5/5!-3^7/7!$$ is a taylor series evaluated at a particular value of x. Find the sumHomework Equations Sum of Infinite series = ##a/1-x## The Attempt at a Solution So, I can't figure out what i would us as the ratio (the thing you multiply...
  26. D

    Proof question: the sum of the reciprocals of the primes diverges

    The gist of the approach I took is that∑1/p = log(e^∑1/p) = log(∏e^1/p) and logx→ ∞ as x→∞. Proof outline: let ∑1/p = s(x). (...SO I can write this easily on tablet) and note that e^s(x) diverges since e^1/p > 1 for any p and the infinite product where every term exceeds 1 is divergent. Then...
  27. J

    MHB Evaluation of Infinite sum of Inverse Trig. Series.

    How can we prove $$\displaystyle \tan^{-1}\left(\frac{4}{7}\right)+\tan^{-1}\left(\frac{4}{19}\right)+\tan^{-1}\left(\frac{4}{39}\right)+\tan^{-1}\left(\frac{4}{67}\right)+...\infty = \frac{\pi}{4}+\cot^{-1}(3)$$ My Trial: First we will calculate $\bf{n^{th}}$ terms of Given Series...
  28. Albert1

    MHB Finding the Sum of $w,x,y,z$ Given $2^w+2^x+2^y+2^z=20.625$

    if $2^w+2^x+2^y+2^z=20.625$ here $w>x>y>z$ and $w,x,y,z \in Z$ find $w+x+y+z$
  29. Albert1

    MHB Find the Min Sum of $m$ and $n$ for $(4^m+4^n)\ mod\ 100=0$

    if $(4^m+4^n)$ mod 100=0 (here $m,n\in N \,\, and \,\,m>n$) please find:$min(m+n)$
  30. J

    Vector Sum of a Standing Wave Confusion

    Hi, I was taught that a standing wave is formed when a progressive wave meets a boundary and is reflected. I was also taught that waves that meet a fixed end, reflect on the opposite side of the axis to the side that they met it at. (I hope that makes sense) If this is true, when the wave is...
  31. F

    Difference of a function and a finite sum

    Hi everybody, I am looking for some help with a problem that has been nagging me for some time now. I'm going to give you the gist of it, but I can provide more details if needed. So, after some calculations I am left with a function of the following form $$ F_L(y) = f(y) -S_L(y)...
  32. adjacent

    C# [C#] Sum of first x natural numbers

    I am writing this in C#. Here is the code. using System; namespace ConsoleApplication3 { class Program { static void Main(string[] args) { int sum = 0; int uservalue; Int32.TryParse(Console.ReadLine(),out uservalue)...
  33. Saitama

    MHB Efficient Methods for Evaluating Complex Sums: A Scientific Approach

    While doing an another problem, I came across the following sum and I have no idea about how one should go about evaluating it. $$\sum_{k=0}^{\infty} (-1)^k\left(\frac{1}{(3k+2)^2}-\frac{1}{(3k+1)^2}\right)$$ Wolfram Alpha gives $-\frac{2\pi^2}{27}$ as the result but I have absolutely no idea...
  34. DreamWeaver

    MHB Finite Binomial Sum: Proving 1 + 1/2 + 1/3 + ... + 1/n

    Show that $$\sum_{j=1}^{j=n}\binom{n}{j} \frac{(-1)^{j+1}}{j} = 1 +\frac{1}{2} +\frac{1}{3} + \cdots +\frac{1}{n}$$
  35. DreamWeaver

    MHB Sum of two inverse tangent functions

    By considering the product of complex numbers: $$z = (2+i)(3+i)$$ Show that $$\tan^{-1}\frac{1}{2} + \tan^{-1}\frac{1}{3} = \frac{\pi}{4}$$
  36. A

    What is the sum of infinite number of zeros?

    dear all, let's consider a length L and divide it into N number of small segments uniformly. Then the length of every segment should be L/N. then we add these segments up, which is L=\sum\frac{L}{N} then we take the limit N→\infty at both sides, this means...
  37. M

    Simple problems regarding sum of IID random variables

    Hi! I'm taking my first course in statistics and am hoping to get some intuition for this set of problems... Suppose I have a bowl of marbles that each weighs m_{marble}=0.01 kg. For each marble I swallow, there is a chance p=0.53 that it adds m_{marble} to my weight, and chance 1-p that...
  38. Joffan

    Sum of arbitrary vertex to midpoint vectors

    I was looking at a homework question posted here requiring proof that the vectors from the vertices of a triangle to the midpoint of the opposite edge sum to zero, and it struck me that there is a more general property: Consider a set of points, \{A_0, A_1, \ldots A_n\}. The midpoint of...
  39. N

    Alternative deduction of sum of sine and cosine

    Hi! Many students know that A\sin(x) + B\cos(x) =\sqrt{A^2+B^2} \sin{(x+\arctan \frac{B}{A})}. I have seen just one deduction of that relation, showed by set up a system of two equations, solving for amplitude and phase shift. Is it possible to deduce the relation in a vectorial way, or in...
  40. R

    Expand an equation - sum and product

    Homework Statement I have been sitting here for the last hour trying to figure it out but I can't seem to be able to find what I'm doing wrong. I need to expand an equation. Homework Equations a2 - a - 3 The Attempt at a Solution a2 - 1a - 3 The product is -3 and the sum -1...
  41. AwesomeTrains

    Partial sum of the harmonic series

    Homework Statement I have to find a natural number N that satisfies this equation: \sum^{N}_{i=1} \frac{1}{i} > 100 Homework Equations I tried finding a close form of the sum but couldn't find anything useful. The Attempt at a Solution Well after trying some numbers in maple I...
  42. anemone

    MHB Find the sum of 5a, 25b, 125c and 625d

    Given $a,\,b,\,c,\,d$ are real numbers such that $a+b+c+d=5$ $2a+4b+8c+16d=7$ $3a+9b+27c+81d=11$ $4a+16b+64c+256d=1$ Evaluate $5a+25b+125c+625d$.
  43. anemone

    MHB Find the sum of all positive integers a

    Find the sum of all positive integers $a$ such that $\sqrt{\sqrt{(a+500)^2-250000}-a}$ is an integer.
  44. G

    Integrating for approximation of a sum

    Homework Statement Find an N so that ##∑^{\infty}_{n=1}\frac{log(n)}{n^2}## is between ##∑^{N}_{n=1}\frac{log (n)}{n^2}## and ##∑^{N}_{n=1}\frac{log(n)}{n^2}+0.005.## Homework Equations Definite integration The Attempt at a Solution I began by taking a definite integral...
  45. anemone

    MHB Evaluating the Sum of $\frac{1}{k_n}$ for $n=1,2,\cdots,1980$

    Let $k_n$ denote the integer closest to $\sqrt{n}$. Evaluate the sum $\dfrac{1}{k_1}+\dfrac{1}{k_2}+\cdots+\dfrac{1}{k_{1980}}$.
  46. G

    Calculating the sum of a sequence

    Homework Statement Compute \sum\frac{4}{(-3)^n}-\frac{3}{3^n} as n begins from 0 and approaches infinity Homework Equations The Attempt at a Solution I'm just getting started on sequences and series, and so far learned about the limit test, comparison test, arithmetic / geometric...
  47. Q

    What Is the Sum of Bond Angles in a Maximized Repulsion Tetrahedron?

    Homework Statement Prove that if bonding-pair repulsions were maximized in CH3X, then the sum of the bond angles would be 450°. Homework Equations In a perfect tetrahedral molecule (e.g. methane), the sum of the bond angles is about 438 degrees (109.5° times 4). The Attempt at a...
  48. U

    Use fourier series to find sum of infinite series

    Homework Statement Find the value of An and given that f(x) = 1 for 0 < x < L/2, find the sum of the infinite series. Homework Equations The Attempt at a Solution The basis is chosen to be ##c_n = \sqrt{\frac{2}{L}}cos (\frac{n\pi }{L}x)## for cosine, and ##s_n = \sqrt{\frac{2}{L}}sin...
  49. Saitama

    MHB Infinite Sum of Powers of x over 1-x^2

    Problem: If $0<x<1$ and $$A_n=\frac{x}{1-x^2}+\frac{x^2}{1-x^4}+\cdots +\frac{x^{2^n}}{1-x^{2^{n+1}}}$$ then find $\displaystyle \lim_{n\rightarrow \infty}A_n$. Attempt: I tried to see if it can be converted to a telescoping series but I had no luck. Then, I tried this: $$\lim_{n\rightarrow...
  50. Saitama

    MHB Evaluating a sum involving binomial coefficients

    Problem: Evaluate $$\mathop{\sum \sum}_{0\leq i<j\leq n} (-1)^{i-j+1}{n\choose i}{n\choose j}$$ Attempt: I wrote the sum as: $$\sum_{j=1}^{n} \sum_{i=0}^{j-1} (-1)^{i-j+1}{n\choose i}{n\choose j}$$ I am not sure how to proceed from here. I tried writing down a few terms but that doesn't seem...
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