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

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5
votes
2answers
193 views

Solving sum of exponentials

Let $x\in\mathbb{R}$, how can $-e^{-x}+e^{x}=a$ be solved? I have already tried to use the sum of exponential formula.
1
vote
1answer
28 views

Discrete Math Induction: $\sum^n_{i=1} \frac1{i(i+1)}$ [duplicate]

For $\sum^n_{i=1} \frac1{i(i+1)}$ Find a formula and proofs that it holds for all n ≥ 1. How would I find the formula for this one that can hold for all n ≥ 1?
-1
votes
2answers
43 views

Discrete Math On Induction proof: $\sum_{i=1}^n n2^n = (n-1)2^{n+1} + 2$ [duplicate]

Show by induction that the following formulas hold. $\sum_{i=1}^n n2ⁿ = (n-1)2^{n+1} + 2$ What did a similar problem to this but this one is a little different. I think is because this one has a ...
-1
votes
1answer
87 views

Use Mathematical Induction to prove that $\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} +…+\frac{1}{n(n+1)}=1-\frac{1}{n+1}$

Use Mathematical induction to prove that for all integers, $n$ is greater than or equal to $1$. I am confused on what to do after I do the the basis step that is using $n$ as $1$. $$\frac{1}{1 \cdot ...
0
votes
0answers
16 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 $$ \int_{0}^{\infty}dt e^{-t}Li_{-k}(-xt) $$ ...
5
votes
3answers
172 views

How do I evaluate this sum(involving the floor function)?

$$ \sum_{i=1}^N\left\lfloor\frac{N}{i}\right\rfloor $$ Is there a closed form expression to the above sum? (Mathematica doesn't give me anything)
0
votes
1answer
18 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} ...
1
vote
2answers
65 views

Sum of positive integers

Find the sum of all positive integers x less than 100 for which (x^2)-81 is a multiple of 100? To do this manually would be too tedious. Also, can someone explain what the latter part of the question ...
1
vote
0answers
22 views

Matlab program of a finite sum that returns the result in a matrix

I am trying to write a simple program for a finite sum from say 0 to 4 of an equation with two variables , say m' and n. each variable takes values form 0 to 4. The result will be in a matrix form and ...
2
votes
3answers
57 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 ...
17
votes
4answers
2k views

Is there a way to denote the calculation $1+2+3+\dots+n$? [duplicate]

Since $n!$ represents $$1\cdot2\cdot3\cdots n,$$ I am wondering if there is a way to represent $$1+2+3+\dots+n?$$ What are some usual notations for the computation of some common sequences? Any other ...
0
votes
1answer
40 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
1answer
39 views

What can I do to this expression to lose the summations?

I'm at the end of a past paper question and need to derive this answer: I am very close and have got to this by doing d/dx to the * equation: What can I do to get rid of these summation signs ...
5
votes
2answers
98 views

The value of $\binom{50}0\binom{50}1+\binom{50}1\binom{50}2+\dots+\binom{50}{49}\binom{50}{50}$ is

The value of $\binom{50}0\binom{50}1+\binom{50}1\binom{50}2+\dots+\binom{50}{49}\binom{50}{50}$ is? I tried this: ...
0
votes
0answers
23 views

`x` increases by 10 everytime`y` occurs`i` times. How do I calculate the sum of `i`?

Can anyone tell me how to calculate this? x increases by 10 everytime y occurs i times. How ...
0
votes
0answers
21 views

Summation of a function with the variable both in the function amd in the upper limit

E is defined as : E = c1 ( a$\rho$ + b$\rho ^{2}$ ) + c2 $\rho$ ( c + d $\sum_{j=0}^{n} (\log{ \frac{R\rho}{j} } ) $ ) + c3 $\rho ^{2}$ a, b, c, d, c1, c2, c3, R are known constants. $\rho$ is the ...
0
votes
2answers
21 views

Telescopic summation

I'm working on this question: Rewrite the following summation using sigma notation and then compute it using the technique of telescoping summation. ...
5
votes
3answers
929 views

Fibonacci trick and proving it. [duplicate]

I am trying to learn Fibonacci tricks and I have one that I can not prove. I know it works because Ive tried it multiple times but I have not a clue how to prove. Here it is: ...
1
vote
2answers
20 views

Compute sum to n

I'm confused as to how to finish answering this question. Compute $$\sum_{i=28}^n (3i^2 - 4i + \frac{5}{7^i}) $$ I end up with $$ 3[\frac{n(n+1)(2n+1)}{6}] - 4[\frac{n(n+1)}{2}] + ...
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 ...
1
vote
3answers
68 views

Calculate $\sum_{k=0}^n k \binom{n}{k} p^k (1-p)^{n-k}$

For $p \in [0,1]$ calculate $$S =\sum_{k=0}^n k \binom{n}{k} p^k (1-p)^{n-k}.$$ Since $$ (1-p)^{n-k} = \sum_{j=0}^{n-k} \binom{n-k}{j} (-p)^j, $$ then $$ S =\sum_{k=0}^n \sum_{j=0}^{n-k} k ...
2
votes
0answers
32 views

Understanding the summation and floor

I am beginner and a novice for sums and math in general. I don't know the steps or techniques to understand how does the right hand side equals the left? Could someone clarify the techniques or steps ...
0
votes
1answer
34 views

Evaluating the following sum

I have no idea how to solve evaluate this integral: $$\lim_{n\to\infty} \frac{1^a + 2^a + \cdots + n^a}{n^{1+a}}, a > -1$$ I want to set this up as some sort of integration since it is a ...
1
vote
2answers
20 views

Find a power series by comparing it to a geometric series?

Find the power series for the following function: $f(x)=\frac{1-x}{x-3}$ centered at x=1. This is what I've done: ...
0
votes
1answer
40 views

Multiplying Sigmas(sums)

I would be grateful if someone please rewrite or expand this please. I have problem multiplying two sigmas ($\sum $) $$ (d(n)-\sum_{k=-\infty}^{\infty} h_k x(n-k)) \times ...
0
votes
1answer
17 views

Probability and Production equation translation

I know that the pi is like a summation except multiplication instead of addition and that P(x) means the probability of, but I'm having trouble putting it all together, esp the $w_i$ such that $w_1, ...
0
votes
2answers
48 views

Algorithm for summing all subsets?

Given a set of integers, is there an algorithm that returns the sum of all the subsets? For example if $s = \{6, 3, -2\}$ then the algorithm returns $28$. I.e: $$(-2) + (3) + (3-2) + 6 + (6-2) + ...
6
votes
1answer
108 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 ...
2
votes
1answer
43 views

Compute summation with a relative error of O(n^-2)

$a(n) = \sum_{i \geq 0} a_i n^{-i}$, how can we compute the value of $a(n)^n$ with a relative error of $O(n^{-2})$?
3
votes
3answers
58 views

How to prove that $\sum_{k=0}^n \binom{2n+1}{2k+1} = 4^{n}$

My question is how to prove that $\sum_{k=0}^n \binom{2n+1}{2k+1} = 4^{n}$ . I'm not good at operating on the sign of sum, so please, try to explain me that as clearly as possible. Thanks :)
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 ...
4
votes
2answers
126 views

Sum of divergent series

I saw a lot of article in Math SE like Why does 1+2+3+⋯=−1/12? and S=1+10+100+100+10000+…=−1/9? How is that and lot of others. Also I saw this one of Ramanujan summation but I do not get the ...
1
vote
1answer
39 views

Triangle inequality frobenius norm

I'm trying to show that the frobenius norm is a norm. however it appears as if triangle inequality isnt met. $$||A+B||_F = \sqrt{\sum_{i,j=1}^n |a_{ij}+b_{ij}|^2} \leq \sqrt{\sum_{i,j=1}^n ...
1
vote
3answers
41 views

Prove by induction:$\sum_{i=0}^n 3^i = \frac {(3^{n+1})-1}{2}$

Prove by induction: $$\sum_{i=0}^n 3^i = \frac {(3^{n+1})-1}{2}$$ Basis: For $n=0$ we have $1 = 1$ Inductive Step: Now this is where I don't know what to do, any kind of help would be much ...
1
vote
1answer
19 views

Convergence of two unusual “nested” sums

I was contemplating convergent sums, trying to think of very unusual or unorthodox sums that might be treatable recursively. Eventually, the following sum occurred to me: $$ \xi = 1 + \frac{ ...
1
vote
0answers
35 views

Is this function monotonically non-decreasing?

I am wondering if the function $L[n]$ defined on $n=0,1,2,\ldots,N$ below is "monotonically" non-decreasing in $n$. I put monotonically in quotes because the function is not continuous and I am not ...
0
votes
2answers
41 views

Inductive proof and summation

The problem asks me to prove by induction that: $$\sum_{i=1}^n i^3 = \frac{n^2(n+1)^2}{4}$$ I've worked through it at least half a dozen times, checked my math fastidiously, can't seem to figure it ...
0
votes
0answers
26 views

Double sum, find upper bound

I have a double sum $$\sum_{i=1}^{\log(n)} \sum_{j=\log(n) - i}^{\log(n)} \left(\frac{1}{2}\right)^i$$ And I'd like to show it's $\mathcal{O}(1)$ i.e. there is a constant $c$ that is an upper bound of ...
2
votes
0answers
37 views

Summation involving Fibonacci numbers

Find: $$ \sum_{n=0}^\infty \sum_{k=0}^n \frac{F_{2k}F_{n-k}}{10^n} $$ where $F_n$ is $n$-th Fibonacci number.
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votes
0answers
27 views

How does one change the top number in a summation?

Sorry I do not know the correct term (I am guessing "upper limit"). Here is what I mean. $$\sum\limits_{i=1}^{\color{red}{17}}\frac{2i}{i+3}$$ The $17$ is what I am talking about as "the top number". ...
0
votes
3answers
31 views

Induction summation proof

Don't want a full answer but can somebody help me in the right direction with this problem. Have to prove using induction $$\forall n \geqslant 2: \sum_{i=1}^{n} \frac{4}{5^{i}} < 1$$
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 ...
1
vote
3answers
60 views

One Binomial Equation $\sum_{i=0}^{z} {n_1 \choose i}{n_2 \choose z-i} = {n_1+n_2 \choose z}$ [duplicate]

I saw one proof using this formula: $$ \sum_{i=0}^{z} {n_1 \choose i}{n_2 \choose z-i} = {n_1+n_2 \choose z}$$ Can anyone help explain it, thank you!
3
votes
2answers
168 views

Is it possible to bound this sum?

I've got a sum: $$\sum_{n=0}^m 9(n+1)10^n$$ And i have big power of 10, like $p=10^{100000}$ And i want to know what is the highest $m$ that this sum will be not greater than $p$? In fact i'm also ...
-1
votes
1answer
35 views

Is there an expression for $(S/k)$ where $S=\sum_{n=1}^\infty n$ and $k \in \mathbb{Z}$?

Given that $S=\sum_{n=1}^\infty n=-1/12$ (for an explanation see this question or this video from Youtube) For example if $k=4$: $(S/4)=1/4+2/4+3/4+1+5/4+6/4+7/4+2+9/4...$ Please edit to improve ...
0
votes
1answer
20 views

Sum of combinations with a condition

Let $m,n,p,q,r$ be non-negative integers, with $0<m\leq n$ and $p+q+r=n$ The identity $\binom{n}{m}=\sum_{x+y+z=m}\binom{p}{x}*\binom{q}{y}*\binom{r}{z}$ holds? I already checked it for m=2, n=5. ...
1
vote
1answer
83 views

Can't find an identy for proving that $ \sum_{k=0}^{i+1} \binom {i+1} k=2^{i+1}$ [duplicate]

$$ \sum_{k=0}^{i+1} \binom {i+1} k$$ I can't find an identity for this summation :( To clarify I'm trying to prove using induction that this sum is equal to $2^{i+1}$, I have my basis and ...
0
votes
1answer
23 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} ...
14
votes
2answers
285 views

How prove this sum $\sum_{n=1}^{\infty}\binom{2n}{n}\frac{(-1)^{n-1}H_{n+1}}{4^n(n+1)}$

show that $$\sum_{n=1}^{\infty}\binom{2n}{n}\dfrac{(-1)^{n-1}H_{n+1}}{4^n(n+1)}=5+4\sqrt{2}\left(\log{\dfrac{2\sqrt{2}}{1+\sqrt{2}}}-1\right)$$ where ...