2
votes
0answers
13 views

What is this sequence of polynomials?

i was trying to calculate the probability of something and i came upon them. i needed to know what this was equal to: $$p_n(x)=\sum_{k_n=k_{n-1}}^{x}....\sum_{k_3=k_2}^{x} \sum_{k_2=k_1}^{x} ...
-1
votes
1answer
23 views

Explanation for sum of sequence

I saw that in a textbook. Could somebody explain how this sum of a sequence was obtained? ⌈n/2⌉+...+⌈n/2⌉+⌈n/2⌉ = ⌈(n+1)/2⌉⌈n/2⌉
2
votes
4answers
83 views

How to find answer to the sum of series $\sum_{n=1}^{\infty}\frac{n}{2^n} $

I have put his on wolfram and obtained answer as follows: $\sum_{n=1}^{\infty}\frac{n}{2^n} = 2$ And the series is convergent too because $\lim_{n\to\infty} \frac {n}{2^n} = 0$ However I am ...
0
votes
2answers
39 views

Calculate the following sequence $\sum_{n=0}^{+\infty }\left ( -\dfrac{1}{4\alpha } \right )^{n}\dfrac{ (2n)!}{n!},\; \alpha >0$

Calculate the following sequence $$\sum_{n=0}^{+\infty }\left ( -\dfrac{1}{4\alpha } \right )^{n}\dfrac{ (2n)!}{n!},\; \alpha >0$$
10
votes
3answers
965 views

Is this already an equation/law that has been found?

So I was messing around with some numbers today and I have found a way to quickly add summations (probably not the first one to discover it but...) this only works when you start at 1 (i.e. ...
2
votes
2answers
53 views

find sum of first 2002 terms

if $\left \{ a_n \right \}$ is sequence of Real Numbers for $n \ge 1$ such that \begin{equation} a_{n+2}=a_{n+1}-a_n \tag{1} \end{equation} \begin{equation} \sum_{n=1}^{999} a_n=1003 \tag{2} ...
0
votes
0answers
17 views

Does this recursion/sequence of iterated infinite sums converge?

Let $n,x\in\mathbb{N}$, $\alpha,\beta,\lambda\in\mathbb{R}^+$, where $\alpha,\beta<1$. Does the following sequence converge (and to what)? $s_0=\alpha n+\lambda$ ...
-2
votes
1answer
48 views

$\sum r 2^r$ series computation [closed]

How do you compute this series? $$ \sum_n n 2^n $$ I tried it the same way as computing the geometric series but I didn't come to a result Hope someone could help Edit The series goes from 0 to ...
0
votes
2answers
53 views

summation of ceil and floor function

I need a closed solution or a faster algorithm for calculating $$ \sum_{k=1}^{n-1} \left\lceil \frac{n}{k}-1 \right\rceil $$ and $$ \sum_{k=1}^{n-1} \left\lfloor \frac{n}{k} \right\rfloor $$ where $ ...
0
votes
0answers
35 views

Precise notation of a sum of a sequence

I need help in rewriting the support of a function $f$ in a more compact or precise way given its upper bound $b$ and lower bound $a$ as \begin{eqnarray} b&=&\max\left( \sum_{n}\alpha_n ...
1
vote
1answer
35 views

Problem Relating to Error in Series

For the following series, find the number of terms required to find the sum with error < 0.005, and find upper and lower bounds for the sum using a much smaller number of terms. ...
1
vote
2answers
20 views

Summation Value of Combination Function

$$\sum_{n=1}^{\infty}\frac{5^n}{n!2^n}$$ I know how to prove that the above series is convergent but I'm not sure how to get the value that it converges to. If it were just the geometric series, I ...
0
votes
1answer
31 views

Convergence of the $\sum_{1}^\infty n^{\frac1n}$

I stumbled upon that question tried some tests like ratio test and ahare test but all gave the limit 1 which is indecisive. Any one with a better approach..
0
votes
0answers
15 views

Borel summation for the series is equal to the laplace transform of the polylogarithm?

let be the series $$ \sum_{n=0}^{\infty}(-1)^{n}n^{k}n! =f(x) $$ then by the Borel summation method would be this sum equal to the Borel transform of $$ \int_{0}^{\infty}dt e^{-t}Li_{-k}(-xt) $$ ...
0
votes
1answer
17 views

Estimating Error of Infinite Series by Finite Series

My book gives the following explanation for finding the error ($ R_{10} $) associated with the sum of the first 10 terms of the following infinite series: $$ (1) \; R_{10}=\sum_{n=1}^{\infty} ...
2
votes
3answers
53 views

Prove that $\lim_{n\to\infty} H_n/n = 0$ ($H_n$ is the $n$-th harmonic number) using certain techniques

I can't seem to use certain methods such as $\varepsilon$-N, L'Hôspital's Rule, Riemann Sums, Integral Test and Divergence Test Contrapositive or Euler's Integral Representation to prove that ...
0
votes
1answer
39 views

How do you find an function equivalent to a summation?

Without intuitively knowing that $\left[1+2+\cdots+n\right]$ is equivalent to $\frac{n^2+n}{2}$, how would I find a function that represents $\sum_{i=1}^{n} i$ ? If possible, I would like to know a ...
1
vote
2answers
43 views

Bounding $\sum_{n=n_1}^\infty x^n (n+1)^2$

I need to upperbound the sum $$\sum_{n=n_1}^\infty x^n (n+1)^2$$ where $0<x<1$ is a parameter. I know it can be done starting from $$\sum_{n=n_1}^\infty x^n (n+1)^2\le \sum_{n=0}^\infty x^n ...
6
votes
1answer
107 views

Find all natural numbers such that $\sum_{k=1}^{n} \frac{n^k}{k!}$ is an integer

Find all natural numbers such that $\sum_{k=1}^{n} \frac{n^k}{k!}$ is an integer. I've tried to bring all fractions under commmon denominator and it didn't helped me much. With guessing I find out ...
0
votes
2answers
21 views

Find a sum of a series

Help me find a sum of this series I tried to excrete as (2/7)^n * 3^(n+2) and use De Lamber indication. It gives me a result 6/7. I checked it in Wolfram Math but the result was 54. Where did I go ...
0
votes
5answers
68 views

How to get this sum

I know the answer to this sum is $$\sum_{k=0}^{i+1} \begin{pmatrix} i+1\\ k\end{pmatrix} = 2^{i+1} $$ because of pascals rule but how do I evaluate the sum to get this? TO clarify I used binomial ...
0
votes
1answer
45 views

How to find this sum

One step away from finishing my proof but not sure how to do this sum: $$\sum_{k=0}^{i+1} \begin{pmatrix} i\\ k-1\end{pmatrix}$$ If it's not easy, if you could explain why, that would be great ...
0
votes
1answer
22 views

Determining value of infinite sum after computing full Fourier Series

I have computed the Full Fourier Series of the function $\phi:[-\pi,\pi] \rightarrow \Bbb{R}$ defined by $\forall x \epsilon[-\pi,\pi], \phi(x)=|\sin(x)|$ to be: $$ \phi(x) = {2\over\pi}+{1\over\pi} ...
2
votes
2answers
31 views

A mixture of AP and GP

A battery loses $10$ mAh of after every hour when in use. The same battery loses $1\%$ of its current amount of charge every hour when not in use. Suppose that the battery is fully charged with ...
2
votes
1answer
67 views

Can someone explain what does this sum mean?

I found a solution to my problem in this thread: How can I (algorithmically) count the number of ways n m-sided dice can add up to a given number? But unfotunately I don't understand the last step. ...
1
vote
2answers
62 views

How can I multiply these sums?

How do I multiply this? Is it even possible given that one sum is infinite? $$ x^{n}\left[\,\sum_{r = 0}^{n}\left(-1\right)^{r}x^{mr}\,\right] \left[\,\sum_{k = 0}^{\infty}{n + k - 1 \choose k} ...
1
vote
3answers
38 views

sum up to nth term with fraction in the power

Is there a formula to express the sum up to the nth term of this: $$ 2^{(1/10)}+2^{(2/10)}+2^{(3/10)}+...+2^{(n/10)}? $$ I am not a mathematician and use computing algorithm but I am looking for a ...
0
votes
2answers
28 views

Compute $\sum_{i=0}^{2n} (-3)^i$ by splitting the series into two parts.

Compute $\sum_{i=0}^{2n} (-3)^i$ by splitting the series into two parts. How do I split it into two parts? All I can tell so far is that the sum is going to be a positive number (probably) because ...
0
votes
2answers
51 views

What is the sum of $1^3q + 2^3q^3 + 3^3q^3 +\cdots+ n^3q^n$?

What is the sum of $1^3*q + 2^3*q^2 + 3^3*q^3 +...+ n^3*q^n ?$
-1
votes
2answers
24 views

Compute the sum $\sum_{i=0}^n 5^{i+1}-5^i$

Compute the sum: $$\sum_{i=0}^n 5^{i+1}-5^i$$ with the hint, "start by writing out (and expanding) the sum." So I did and got $$4 + 20 + 100...$$ with the appearance of going to infinity. Is ...
0
votes
1answer
15 views

What steps and properties are involved in this Summation simplification?

I have reference material that shows how to get from step 2 to 3 by recognizing the harmonic series. But how do they get from step 1 to 2?
0
votes
1answer
42 views

Prove there is a subsequence $(a_{nk})_{n=1}^\infty$ such that $\Sigma^{\infty}_{k=1} a_{nk}$ converges.

Hey everyone this was give as a practice problem and i'm having trouble, any help is appreciated Let $(a_n)_{n=1}^\infty$ be a sequence such that $\displaystyle \lim_{n \rightarrow \infty} {a_n} = ...
0
votes
1answer
59 views

Inequality Question about Converging Sum

This is from the UPenn prelim questions. http://hans.math.upenn.edu/amcs/AMCS/prelims/prelim_review.pdf (I asked the question before, and there was no answer, so I am asking it again.. I'm not sure ...
2
votes
4answers
258 views

Why is my series wrong?

Why is this series wrong and how does it differ from this other one? We had to find the general term for the series: $ 1/3+2/9+1/27+2/81+1/243+2/729+\ldots $ where the index begins at $n=1$ So I came ...
0
votes
0answers
32 views

Approximation for the logarithm of a summatory

I would like to find an approximation for: $$ \log \left(\sum_{i=1}^{N} a_i\exp(-b_i^2)\exp(-c_i^2)\right) $$ with $$ a_i = \frac{1}{\sqrt{(e^2 + e_i^2)(g^2 + g_i^2)}} \\ b_i = \frac{b-d_i}{2(e^2 + ...
2
votes
0answers
33 views

Sum of Binomials times Logarithms

Is there a closed-form expression or a very good approximation for $$ \sum_{i=0}^n \binom{n}{i} \log (i+1) \,? $$ If the summands alternate, then there is a very close approximation, yet it feels ...
1
vote
4answers
80 views

Evaluating $\lim_{n\to\infty} \left({1\over1\cdot2\cdot3}+{1\over2\cdot3\cdot4}+\cdots+{1\over(n-1)\cdot n\cdot(n+1)}\right)$

The original question was to find $L=\displaystyle\lim_{n\to\infty}\sum_{k=1}^na_k$ where $a_n=\displaystyle{n\over(1+2+\cdots+(n-1))(1+2+\cdots+n)}$, which I managed to get down to evaluating the ...
1
vote
2answers
185 views

Is $1 + 2 + 3 + \dots = -\frac{1}{12}$ really true? [duplicate]

I've read this strange result of the sum of all positive integers being $-\frac{1}{12}$. Is it really true? Does this also mean this is true? $$\sum_{n=1}^k n = \frac{k\cdot(k+1)}{2}$$ ...
7
votes
3answers
428 views

How to find the sum of $i(i+1)\cdots(i+k)$ for fixed $k$ between $i = 1$ and $n$?

I learned that $$\sum \limits_{i=1}^n i(i+1) = \frac{n(n+1)(n+2)}{3}$$ or in general $$\sum \limits_{i = 1}^n i(i+1)(i+2) \dots (i + k) = \frac{n(n+1)\dots (n+k+1)}{k+2}$$ From a mathematical ...
0
votes
1answer
29 views

Is $f$ integrable in $L(X,\mathcal{X},\mu)$

Is $f$ integrable $L(X,\mathcal{X},\mu)$ $\mu(E)=\sum_{n\in E\cap\mathbb{N}} |n^2+n-6|$ $f:\mathbb{R}\rightarrow \mathbb{R_+}\cup\infty$ $f=(x-2)^{-4}$
0
votes
1answer
41 views

Prove that if $\sum_{n=1}^\infty a_{n}$ is absolutely convergent, then $|\sum_{n=1}^\infty a_{n}| \leq | \sum_{n=1}^\infty |a_{n}|$

Hey everyone this was given as a practice problem for my first year calculus class and it really giving me a headache, any help is appreciated! Prove that if $\sum_{n=1}^\infty a_{n}$ is absoultley ...
1
vote
3answers
106 views

Multiplication of infinite series

Why multiplication of finite sums $(\sum_{i=0}^n a_i)(\sum_{i=0}^n b_i)=\sum_{i=0}^n (\sum_{j=0}^ia_jb_{i-j})$ (EDIT: This assumption was shown to be false) does not work in infinite case? I have ...
11
votes
2answers
681 views

Why is this allowed? (“Fourier's Trick”; finding the coefficients in a Fourier Series)

In my textbook (Introduction to Electrodynamics, D. Griffiths), we derive the equation for some strange potential function. Eventually, we get to this (for $n \in \mathbb{Z}^+$): $$ V_0(y) = ...
3
votes
1answer
87 views

Proof that $0 < \lim(a_n/b_n) < \infty$ implies convergence/divergence of $a_n$ and $b_n$

Suppose that $a_n, b_n > 0$ for all $n$. Prove that if $0 < \lim(a_n/b_n) < \infty$, then $\sum a_n$ and $\sum b_n$ either both converge or both diverge. I'm a bit unsure on how to proceed ...
1
vote
2answers
72 views

How prove this series $\sum_{n=1}^{\infty}\frac{e^n\cdot n!}{n^n}$ divergent

show that this series $$\sum_{n=1}^{\infty}\dfrac{e^n\cdot n!}{n^n}$$ divergent My try: since $$u_{n}=\dfrac{e^n\cdot n!}{n^n},\Longrightarrow ...
1
vote
1answer
27 views

Series Sum up to N terms

I have been trying to find the sum of a series given by $ t(n) = \frac{1}{2^n-1}$, up to N terms. All I could do is to see that the difference of the successive denominators form a GP. Kindly help me ...
7
votes
0answers
132 views

A Ramanujan-like summation: is it correct? Is it extensible?

I'm still exercising with summation-procedures which I try to make correct Ramanujan-summations. Looking at the series $$ s(1/2,2) = (1/2)+(1/2)^4+(1/2)^9+(1/2)^{16}+... $$ and more general $$ s(b,p) ...
0
votes
0answers
38 views

Way to split up product of summation

If I have $\sum_{n=1}^{\infty}f(x)g(x)$, is there any way to split this up? Thanks.
0
votes
3answers
56 views

Summation of 1/k [duplicate]

What is summation of 1/k where n ranges from 1 to n. I need the general formula for the summation. I know the series tends to infinity when k tends to infinity . But upto n terms there must be a ...
0
votes
1answer
37 views

Evaluate the following sum

I need explanation how they get to the following equation from left to right considering the partions are $\frac{n}{n}$ $$\sum_{i=1}^n ...