7
votes
4answers
67 views

A closed form of $\sum_{k=0}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right)$

I am looking for a closed form of the following series \begin{equation} \mathcal{I}=\sum_{k=1}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right) \end{equation} I have no idea how to ...
1
vote
4answers
146 views

Find whether the following series converges or diverges $\sum_{n=1}^{\infty}\frac{\ln n }{\sqrt{n}}$

Looking for a witty answer. I can see that the given series converges by AST. All Ideas Appreciated
1
vote
2answers
45 views

Matrix Inversion Test ( Sum of Matrix series)

Friends,I have a set of matrices of dimension $3\times3$ called $A_i$. , Following are the given conditions a) each $A_i$ is non invertible except $A_0$ because their determinant is zero. b) ...
1
vote
1answer
26 views

Sum of nth powers and generalized polynomial sum

So this is a 2-part question (both parts I believe are closely related): How exactly does on express the sum $$\sum_{i=0}^{k}{i^n} = Q(n,k)$$ in a closed form For arbitrary positive integers ...
13
votes
2answers
497 views

Intuitive ways to get formula of cubic sum

Is there an intuitive way to get cubic sum? From this post: combination of quadratic and cubic series and Wikipedia: Faulhaber formula, I get $$1^3 + 2^3 + \dots + n^3 = \frac{n^2(n+1)^2}{4}$$ I think ...
1
vote
1answer
27 views

find a $B_{n,j}$ such that $|A_{n,j}-L_j| \leq B_{n,j}$ $\forall n,j$ and $\sum_{j=0}^{\infty}B_{n,j}$ converges

We have $A_{n,j}= 3(-1)^j2^{n-j+1}\frac{(2(n-j)-4)!}{(n-j)!(n-j-2)!}\binom{j+2}{2}\frac{n^\frac{5}{2}}{8^n}$ and $L_j=(-\frac{1}{8})^j\binom{j+2}{2}\frac{3}{8\sqrt{\pi}}$ So I know $\lim_{n \to ...
7
votes
3answers
171 views

Calculate $\frac{1}{5^1}+\frac{3}{5^3}+\frac{5}{5^5}+\frac{7}{5^7}+\frac{9}{5^9}+\cdots$

I'm an eight-grader and I need help to answer this math problem. Problem: Calculate $$\frac{1}{5^1}+\frac{3}{5^3}+\frac{5}{5^5}+\frac{7}{5^7}+\frac{9}{5^9}+\cdots$$ This one is very hard for me. It ...
1
vote
1answer
72 views

combination of quadratic and cubic series

I'm an eight-grader and I need help to answer this math problem (homework). Problem: Calculate $$\frac{1^2+2^2+3^2+4^2+...+1000^2}{1^3+2^3+3^3+4^3+...+1000^3}$$ Attempt: I know how to calculate ...
3
votes
1answer
43 views

Measuring sums of complex alternating series

Suppose we have functions $$f(x) = \sqrt{x}, \space g(f) = \frac{df}{dx}+\frac{d^2f}{dx^2}+\frac{d^3f}{dx^3}\space ...$$ Applying function f(x) to g(f) we get: $$g(f(x))=\frac{1}{2}x^{-\frac{1}{2}} - ...
2
votes
2answers
48 views

is there a generating function for $H_{2n}$?

I have been wondering if anyone knows if there is a generating function for harmonic series of the form $H_{2n}$?. That is, we are familiar with ...
3
votes
1answer
44 views

investigate $\sum\limits_{n\ge1}{\frac{(-1)^n}{n^\alpha \ln n}}$

I need to investigate the series (Hence, when the series converges and when the series converges absolutely depending on $\alpha$). $$\sum\limits_{n\ge2}{\frac{(-1)^n}{n^\alpha \ln n}}$$ For ...
3
votes
2answers
82 views

Prove by induction that $A_k = \sum\limits_{n=2k}^{3k}\binom{3k}{n}\cdot{(\frac{60}{100})}^n\cdot{(\frac{40}{100})}^{3k-n}$ is decreasing

I want to prove that the following sequence is monotonously decreasing: $A_k = \sum\limits_{n=2k}^{3k}\binom{3k}{n}\cdot{(\frac{60}{100})}^n\cdot{(\frac{40}{100})}^{3k-n}$ I think it should be ...
3
votes
3answers
72 views

Closed form for the sum: $\sum_{n=1}^{\infty}\frac{1}{n(n + 1/3)}$ [duplicate]

I tried using partial fractions to compute the sum of the series $$ \sum_{n=1}^{\infty}\frac{1}{n(n + 1/3)} $$ Another technique is to turn this series into a definite integral of 0 to 1. but do not ...
2
votes
1answer
46 views

Identity with binomials [duplicate]

Does there exist a closed formula for $$\underset{n=1}{\overset{N-1}{\sum}}\dbinom{N+n}{n}?$$ I've searching on wikipedia but I haven't found this kind of sum.
2
votes
3answers
42 views

Series question with logarithms

I want to know how to check the divergence of following sum: $\sum_{k=0}^\infty \frac{1}{\sqrt[n]{\log n}}$ I tried to use this result: $ \lim_{n \rightarrow \infty} \frac{1}{\sqrt[n]{\log n}}=1 ...
7
votes
2answers
158 views

Prove that $\displaystyle{\sum_{n=1}^{\infty}}(-1)^{n-1} \dfrac{H_n}{n} = \dfrac{\pi^2}{12} - \dfrac{1}{2}\ln^2 2$

We know that $H_n = \sum_{j=1}^{n}{1 \over j}$. Article in The Sum of Certain Series Related To Harmonic Numbers of Omran Kolba, we have proof of this identity which involves some advanced concepts. ...
4
votes
2answers
148 views

Combination of quadratic and arithmetic series

Problem: Calculate $\dfrac{1^2+2^2+3^2+4^2+\cdots+23333330^2}{1+2+3+4+\cdots+23333330}$. Attempt: I know the denominator is arithmetic series and equals ...
0
votes
0answers
19 views

Sum of squares of series of boolean variables

I am going to simplify the following series: $$\sum^4_{v=1} \left(1 - \sum^4_{i=1} x_{v,i}\right)^2 + \sum^4_{i=1} \left(1 - \sum^4_{v=1} x_{v,i}\right)^2$$ Since $x_{i,j}$ is a boolean variable, ...
0
votes
2answers
36 views

Sum of this series.

I tried manipulating it to get it into a binomial expansion of two known terms, but i seemingly failed. Please help me out. $$S=\displaystyle\sum_{r=0}^{12} \binom{12}{r} \cos \frac {r\pi}{6}$$
12
votes
3answers
233 views

How to prove that $ \sum_{n=1}^{\infty}n\prod_{k=1}^{n}\frac{1}{1+ka} = \frac{1}{a} $?

Mathematica tells me that $$ \sum_{n=1}^{\infty}n\prod_{k=1}^{n}\frac{1}{1+ka} = \frac{1}{a} $$ I could prove it for $a\rightarrow 0$, $a=1$ and $a\rightarrow \infty$, but could not find a general ...
4
votes
2answers
74 views

Show that the following series is less than $4 \pi^2 / 3$.

Show: For any $k = 0,1,2,...$, $$ \sum_{i=0}^{i=k} \frac{(k+1)^2}{(i+1)^2 (k-i+1)^2} \leq \frac{4 \pi^2}{3}. $$
1
vote
3answers
61 views

Can you show that the LHS equals the RHS in this equation, by showing how I can get the expression on the RHS?

$$ \frac{1^2+2^2+...+(n-1)^2}{n^3} = \frac{(n-1)n(2n-1)}{6n^3} $$ Can someone show me step by step how I can transform the LHS to the RHS? If possible, using high school-level math. I have now ...
0
votes
0answers
44 views

Intuition behind summation [duplicate]

$$\sum_{n=1}^{\infty}n = -1/12$$ Could someone give me an intuitive explanation of why this is true? I just completed Calc 2 and finished a unit on series, and according to the stuff we learn't the ...
5
votes
1answer
62 views

How many decimal representations are possible for the number 1

I know that there at least two $0.\overline{9}$ and 1 Is there a neat and more concrete way to understand this problem.
11
votes
1answer
80 views

Convergence of $\sum_{n=1}^{+\infty}\frac{(-1)^{f(n)}}{n}$ where $f(n)$ is the number of prime divisors

Let $f(n)$ be the number of prime divisors of a number $n$ counted with their multiplicities. Show that the series $\sum_{n=1}^{+\infty}\frac{(-1)^{f(n)}}{n}$ converges and has sum $0$. Attempt ...
1
vote
1answer
116 views

Find the sum of the series $\sum \limits_{n=3}^{\infty} \dfrac{1}{n^5-5n^3+4n}$

Feel free to skip obvious steps, or use a calculator when required. I just want to understand the theme of the solution. Any help is appreciated EDIT : We can write$$ \dfrac{1}{n^5-5n^3+4n} = ...
0
votes
2answers
31 views

If a sequence $\{a_n\}$ satisfies the Inequality $a_{n+1} < ka_{n}$, then show that $ \lim\limits_{n \to \infty} a_n =0$ where $0< k , a_n< 1$

I know one solution. Consider $\sum a_n$ Then use ratio test to show that the series converges, hence the sequence. Any other Ideass !
1
vote
3answers
63 views

Find all values of $c$ for which the following series converges $\sum_{n=1}^{\infty} \left(\dfrac{c}{n}-\dfrac{1}{n+1}\right)$

I know that the series in question converges when $c=1$, but I have no concrete way to find all such values of $c$ for which this is true.
3
votes
1answer
52 views

Give example of a series $\sum a_n$ such that the series is conditionally convergent. and $\sum na_n$ is convergent

I tried all the conditionally convergent series I know, I found $\sum na_n$ to be diverging for all of them. But I am sure the question is correct
7
votes
1answer
203 views

Series $\sum \frac{1}{n^2\sin^3n}$

Question : Show that series $\sum \cfrac{1}{n^{2}\sin^{3}n}$ is divergent. Hint: Show that $$\sum \frac{1}{n|\sin(n)|}$$ is divergent. I am interested in other possible proofs for this question. ...
0
votes
1answer
60 views

Concerning the sum $\sum_{n = 1}^\infty \sin nx$

I recently came across this question and I posted an answer. It has been pointed out that my answer is incorrect. I cannot work out what is wrong with my reasoning. The answer I gave corresponds with ...
3
votes
6answers
112 views

How can I show that $\sum_{k=0}^{\infty} \frac{k-1}{2^k} = 0$?

I'm studying the algorithms book and I have a doubt. I don't know how can I prove this summation: $$\sum_{k=0}^{\infty} \frac{(k-1)}{2^k} = 0$$
2
votes
0answers
95 views

Find the limit of $\sum \frac{1}{log^n(n)}$

Working on convergence and divergence of infinite series, I recently focused my attention on the summation $$\displaystyle\sum\limits_{n=2}^{\infty} \frac{1}{log^n(n)}$$ While proving the convergence ...
1
vote
1answer
40 views

Algebraic manipulation of floors and ceilings

I am trying to solve the summation $$ \sum_{n=3}^{\infty} \frac{1}{2^n} \sum_{i=3}^{n} \left\lceil\frac{i-2}{2}\right\rceil $$ I will list some of the simplifications that I've found so far, and ...
4
votes
3answers
164 views

Infinite Sum of algebraic expression

Prove that $$\sum_{i=1}^{\infty} \frac{1}{i(2i+3)} = \frac89 -\frac23\ln2$$ I tried using integration but failed miserably. Hints please.
1
vote
0answers
83 views

What is the sum of Psi/Digamma-function of consecutive arguments? Is there a closed form?

In a consideration of summation of a series $$ s = a_0 + a_1 + a_2 + \cdots \tag 1$$ with $$\lim_{k \to \infty} a_k=0$$ but slowly decreasing, the coefficients $a_k$ are somehow related to $1/k^2$ ...
23
votes
1answer
326 views
+50

Evaluating the sum $\lim_{n\to \infty}\sqrt[2]{2+\sqrt[3]{2+\sqrt[4]{2+\cdots+\sqrt[n]{2}}}}$

The following nested radical $$\lim_{n\to \infty}\sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}$$ is known to converge to 2. We can consider a similar nested radical where the degree of the radicals increases: ...
2
votes
0answers
34 views

statements about summation

Could you help me prove this statements about summation? I know that the second prove is easy of be written, but can I put that summation before cos(theta) and sin(theta)? yes. But why? Do you ...
1
vote
1answer
26 views

Discrete math: Sum of Geometric series on a problem - Notes make little sense.

I've been reading a PDF of slides from my Discrete Math I professor. The title is Sums, Products and Asymptotic Estimations. He gives us a problem to fire off the lecture, which is the following: ...
1
vote
0answers
28 views

Weighted sum of $\cos(nx)$ series

This is a follow up question to Prove $\frac{1}{2} + \cos(x) + \cos(2x) + \dots+ \cos(nx) = \frac{\sin(n+\frac{1}{2})x}{2\sin(\frac{1}{2}x)}$ for $x \neq 0, \pm 2\pi, \pm 4\pi,\dots$ I am looking ...
2
votes
1answer
80 views

How Find the sum $\sum_{n=2}^{\infty}\left(\sum_{j=1}^{n}\frac{1}{(2j-1)^2} \binom{2n}{n}\frac{1}{2^{2n}(2n+1)}\right)$

Prove or disprove $$I=\sum_{n=2}^{\infty}\left(1+\dfrac{1}{3^2}+\dfrac{1}{5^2}+\cdots+\dfrac{1}{(2n-1)^2}\right) \binom{2n}{n}\dfrac{1}{2^{2n}(2n+1)}=\dfrac{\pi^3}{48}-\dfrac{1}{6}$$ My ...
0
votes
2answers
21 views

Sequences and Series - neither an AP or a GP, how do I solve it

Find the sum of the first n terms of a series given $T_r = 2^r +2r - 1$ I've worked out the first six terms and found them to be; $3, 7, 13, 23,41$ and $75.$ Working out their differences we get ...
0
votes
0answers
42 views

Gosper summable

I'd like to know why the following is NOT gosper summable: $$\sum_{k\in \Bbb{Z}} \frac{p(k)}{\prod_{j=0}^{m-1}(k+a+j)}$$ where $m>0, m\in\Bbb{Z}$ and $p(k)$ is a polynomial of degree $k=m-1$.
3
votes
0answers
56 views

Question about factorial function [duplicate]

Show that $$n!=1+\left(1−{1 \over 1!}\right)n+\left(1−{1 \over 1!}+ {1 \over 2!}\right)n(n−1)+\cdots$$ I can't figure out how this can be solved . I tried to use binomial theorem but I couldn't prove ...
3
votes
2answers
64 views

Simple factorial function question

Show that $n!=1+(1-1/1!)n+(1-1/1!+1/2!)n(n-1)+....$ I can't figure out how this can be solved . I tried to use binomial theorem but I couldn't prove it . Any help will be greatly appreciated.
12
votes
4answers
245 views

Show $\sum_{k=1}^{\infty}\left(\frac{1+\sin(k)}{2}\right)^k$ diverges

Show$$\sum_{k=1}^{\infty}\left(\frac{1+\sin(k)}{2}\right)^k$$diverges. Just going down the list, the following tests don't work (or I failed at using them correctly) because: $\lim ...
1
vote
4answers
93 views

Prove the sum $\sum_{n=1}^\infty \frac{\arctan{n}}{n}$ diverges.

I must prove, that sum diverges, but... $$\sum_{n=1}^\infty \frac{\arctan{n}}{n}$$ $$\lim_{n \to \infty} \frac{\arctan{n}}{n} = \frac{\pi/2}{\infty} = 0$$ $$\lim_{n \to \infty} \frac{ ...
2
votes
5answers
125 views

Find a closed form for $\sum_{k=0}^{n} k^3$ [duplicate]

Find a closed form for $\sum_{k=0}^{n} k^3$. I would appreciate ideas for approaching questions like this in general as well. Thanks.
0
votes
2answers
43 views

double summation problem $\sum^5_{i=1}i \times \sum^5_{j=1}j =…$ please check

(I) $\sum^5_{i=1}i \times \sum^5_{j=1}j = 1 \times (1) +1 \times (2) + \cdots +1\times (5) +2\times (1)+2\times (2) +\cdots + 2\times (5) + 3\times (1) + 3\times (2) + \cdots +3\times (5) + 4\times ...
1
vote
3answers
24 views

Positive positive negative negative Series

What is the simplest series that alternates in order $+,+,-,-,+,+,-,- \dots$ Specifically I want to make a Riemann sum for something, but it has this reoccurent pattern I haven't previously ...