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

learn more… | top users | synonyms

0
votes
1answer
27 views

How to simplify this triple summation

I am trying to calculate the following summation by n : $$\begin{align} \sum_{i=1}^n \sum_{j=i}^n \sum_{k=i}^1 &= \sum_{i=1}^n \sum_{j=i}^n (j-i+1) \\ &= \sum_{i=1}^n \left( ...
1
vote
2answers
29 views

Conversion from sum of product to product of sum

I do not understand how did he convert from this to this. Source :http://cs229.stanford.edu/notes/cs229-notes1.pdf Page 18
0
votes
1answer
16 views

Sum of a “geometric series” with an extra term (a[i] = 2a[i-1] + i)

For the series defined by a[0] = 0, a[i] = 2a[i-1] + i, WolframAlpha gives me the sum as ...
3
votes
1answer
38 views

Can two distinct sets of $N$ non-negative numbers have the same sum and sum of squares?

Suppose I have a set of $N$ non-negative numbers that sum to $A$. The sum of squares of these $N$ non-negative numbers sum to $B$. Here's the question: can there be a different set of $N$ ...
0
votes
0answers
15 views

Show that autocorrelation of a line is a line

Autocorrelation for a time series is defined as: $$\frac{\sum_{t=1}^{n-v}{(X_t - \bar{X})(X_{t+v} - \bar{X}) }} {\sum_{t=1}^{n}{(X_t - \bar{X})^2}} $$ Where $v$ is the lag $1..m,$ $m < n$ How ...
7
votes
2answers
68 views

Summation of the reciprocals of the product of consecutive integers

It is well known that there is a closed formula for: $$\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \cdots + \frac{1}{(n)(n + 1)}$$ And likewise for: $$\frac{1}{1 \cdot 2 \cdot 3} + \frac{1}{2 \cdot ...
3
votes
1answer
23 views

Simplifying a Taylor polynomial that involves Stirling numbers of the second kind

I am currently trying to evaluate a complicated function $f$ at a point $x+a$ using a high order Taylor polynomial about the point $a$. The polynomial has the standard form: $$\sum_{n=1}^k ...
1
vote
1answer
58 views

Find the limit $\lim\limits_{n\to \infty}\sum\limits_{k=1}^{n}\left(\frac{k}{n^2}\right)^{\frac{k}{n^2}+1}$

Find the limit $$\lim_{n\to \infty}\sum_{k=1}^{n}\left(\frac{k}{n^2}\right )^{\frac{k}{n^2}+1}$$ Have no idea.
1
vote
0answers
26 views

Proof of the binomial theorem through Dirichlet convolution?

Here I gave a proof for $\sum_{k=0}^n\binom nk(-1)^k=0$ based on the fact that $\mu*1=\varepsilon$ (the Dirichlet identity). I am wondering if using a similar technique we can prove that ...
0
votes
0answers
27 views

Moebius Identity

Is there alternative proof of Moebius identity i.e. sum of moebius function over divisor of n is zero than as suggested n page: ...
0
votes
0answers
13 views

Dirichlet product is associative

Is there alternative proof of fact: Dirichlet product on arithmetic function is associative than given in Dirichlet's product with number theoretic functions
7
votes
0answers
34 views

Does $S(n)$ contain infinite many primes? [duplicate]

Denote $p_j := j\text{th prime}$ and $S(n)\:=\sum_{j=1}^n p_j$ (The sum of the first $n$ primes). Is it known whether $S(n)$ is prime for infinite many $n$? OEIS gives the sum of the prime ...
0
votes
1answer
22 views

Sum interpretation

Is the interpretation of $$ \sum_{1 \leq j-1 \leq n}a_{j-1} $$ in that stile $$ \sum_{j-1=n}^{n}a_{j-1} $$ correct ? Can somebody give me a numerical example of that sum, please ?
5
votes
6answers
109 views

Prove a lower bound for $\sum_{i=1}^n i^2$

Prove that $$\sum_{i=1}^n i^2 \geq \frac{n^3}{3}$$ for all $n \geq 1.$ What I know: I know the basic format of how to make a proof with the basis and inductive step but I am unsure of how to ...
2
votes
2answers
65 views

Prove that $\sum_{t=1}^{p-1} \frac{t^2-1}{t^2+1} \equiv 0 \pmod p$

I'm trying to prove the statement $\sum_{t=1}^{p-1} \frac{t^2-1}{t^2+1} \equiv 0 \mod p$ and I don't really know where to start. Obviously $\sum_{t=1}^{p-1} \frac{t^2-1}{t^2+1} = 2\sum_{t=1}^{(p-1)/2} ...
2
votes
1answer
41 views

Use the 'rule of sum to prove that $\sum_{k=0}^n 2^k=2^{n+1}-1$.'

I'm attempting a combinatorics problem that asks to use the 'rule of sum to prove that $\sum_{k=0}^n 2^k=2^{n+1}-1$.' The rule of sum says that 'if $S=\cup_{i=1}^t S_i$ is a union of disjoint sets ...
0
votes
0answers
39 views

is it possible to evaluate any definite integral using the definition of the definite integral?

I was evaluating definite integrals using the fundamental theorem, however, out of curiosity, I wanted to see if it was possible to evaluate the following, using the definition of the definite ...
1
vote
2answers
68 views

How to solve $\sum_{k=1}^n\frac{k}{n^2+k}$?

Can someone show me what is wrong with the expression I got for evaluating $\sum_{k=1}^n\frac{k}{n^2+k}$? Steps: $\sum_{k=1}^n\frac{k}{n^2+k} = \frac{\sum_{k=1}^nk}{\sum_{k=1}^{n}n^2+k} = ...
1
vote
2answers
35 views

Convergent Summation Proof

I'll preface and apologize for the somewhat lengthy intro. Alright, so recently I've been self-studying summation and found interest in convergent series. I was looking at some fairly common series ...
12
votes
4answers
606 views

Can we add an uncountable number of positive elements, and can this sum be finite?

Can we add an uncountable number of positive elements, and can this sum be finite? I always have trouble understanding mathematical operations when dealing with an uncountable number of elements. ...
5
votes
2answers
124 views

What is the sum of this series? $\sum_{n=1}^\infty\frac{\zeta(1-n)(-1)^{n+1}}{2^{n-1}}$

I want to know what is the sum of this series: $$\sum_{n=1}^\infty\frac{\zeta(1-n)(-1)^{n+1}}{2^{n-1}}$$ $B_n$ are the Bernoulli numbers. Mathematica does not help.
0
votes
1answer
51 views

What is the sum of this series? $\sum_{n=1}^\infty\frac{2^n (-1)^{n+1} B_n}{n}$ [duplicate]

I want to know what is the sum of this series: $$\sum_{n=1}^\infty\frac{2^n (-1)^{n+1} B_n}{n}$$ Mathematica does not help.
0
votes
2answers
43 views

Can absolute value functions be moved like this?

If I have an expression that looks like $|x-a_1| + |x-a_2| + |x-a_3| + ... + |x-a_n|$ Is it the same as doing $|nx - \sum_{i=1}^{n}a_i|$
2
votes
0answers
62 views

The sum of $\frac{k^{2}}{k-1}$ from 6 to 12

Can you evaluate this sum by using the properties of the sigma notation ? Or I must develop this and evaluate them one by one ?
2
votes
1answer
53 views

How many terms required in $e =\sum^∞_{k=0}{1\over k!}$ to give $e$ with an error of at most ${6\over 10}$ unit in the $20$th decimal place?

How many terms are required in the series $e =\sum^∞_{k=0}{1\over k!}$ to give $e$ with an error of at most ${6\over 10}$ unit in the $20$th decimal place? Here is what I have: $$e\approx ...
0
votes
0answers
43 views

tricky derivative with logarithm of sum

I'm having trouble understanding the solution of a limit. It involves a formula for measuring certainty of a discrete probability distribution. Given a set of values $p_j$ which sum up to 1, find the ...
1
vote
1answer
151 views

Why is $1+2+3+\cdots = 0 $? [duplicate]

I had seen this result a while back in a Numberphile video: $1+2+3+\cdots = -\frac{1}{12}$ I was trying to prove the same result using a different method when I accidently proved that the sum was ...
-5
votes
0answers
31 views

Summation from $2^0 log_2(n)$ to $2^{n-1} log_2(1)$ [closed]

What is the sum: $2^0 log_2(n) + 2^1log_2(n-1) + 2^2log_2(n-2) + ... + 2^{n-2}log_2(2) + 2^{n-1}log_2(1)$?
-5
votes
6answers
59 views

What is $1+2+4+8+16+…+2^n$? [duplicate]

What is the result of: summation from one, two, four, eight until $n$ power of two? Thank you!
2
votes
2answers
62 views

How can I solve $\sum_{n=1}^{\infty} \frac{8}{(4n-1)(4n-3)}$ step by step? [closed]

$$\sum_{n=1}^{\infty} \frac{8}{(4n-1)(4n-3)}$$ First off I know the answer to this question, but I do not know how to actually get the answer I do know some basic sums like that one. Can someone ...
2
votes
2answers
86 views

If $(a+b)^n=\sum_{k=0}^{n}{n\choose k}a^{n-k}b^kc_k$, then $c_k=1$?

Be advised this is a real soft question: If $$(a+b)^n=\sum_{k=0}^{n}{n\choose k}a^{n-k}b^kc_k$$ Assuming $abc \neq 0$ must we have the following condition? $$c_k=1$$ for all $0 \leq k \leq n$ How do ...
1
vote
4answers
39 views

Solving this Recurrence Relation in terms of previous values.

What will be the value of $X(n)$ and $Y(n)$ in terms of given $n,X(0),Y(0)$. $$ X(n) = X(n-1) + Y(n-1) \\ Y(n) = 2X(n-1) + Y(n-1) $$
4
votes
3answers
75 views

Sum involving zeta functions

Find closed form of the following - $$ \displaystyle \sum_{n=2}^{\infty}{\left(\frac{(n-1)\zeta(n)}{4n-1}\right)} $$ I don't know how to approach to it - Using the integral definition? I cannot use ...
0
votes
2answers
28 views

Simplifying with Summation

This is a problem out of my statistics book but my issue is simplifying from Step 3 to Step 4 below: Step 1: var X=$\sum\:p_i\:(x_i-E[X])^2$ Step 2: var X=$\sum\:p_i[x_i^2+E[X]^2-2x_iE[X]]$ Step ...
3
votes
1answer
37 views

bounding a sum using a definite integral

Conjecture. Let $1<p<\infty$. Then there exists $C\in(0,\infty)$ such that for any $k\in\mathbb{Z}^+$ we have \begin{equation}\sum_{n=1}^k(k+1-n)^{-\frac{p}{p+1}}n^{-\frac{1}{p+1}}\leq ...
5
votes
2answers
165 views

What is the integer part of $\sum_{i=2}^{9999} \frac {1}{\sqrt i}?$

What is the integer part of: $$\sum_{i=2}^{9999} \frac {1}{\sqrt {i}}$$ A short but tricky problem. Any help is welcome.
0
votes
0answers
19 views

Does this theorem concerning upper and lower bnound of a monotone decreasing function have a formal name?

This is the theorem: Let $g$ be a monotone decreasing function and let $a,b \in \mathbb{N}$. Then the following holds true: $$\int_{a}^{b+1}g(x)dx \overset{(i)}{\leq} ...
3
votes
3answers
51 views

Does $\lim\limits_{n \to +∞} \sum_{k=1}^n \frac{n\cdot \ln (k)}{n^2+k^2}$ diverge?

Does the limit of this summation diverge? $$\lim\limits_{n \to +∞} \sum_{k=1}^n \frac{n\cdot \ln (k)}{n^2+k^2}$$ Thanks!
1
vote
2answers
35 views

Question about summations with an unknown lower limit

I am unsure of how to proceed about finding the solution to this problem. $$\sum_{i=6}^8(\sum_{j=i}^8 (j+1)^2)$$ Obviously the last step is not to difficult, but the fact that the lower limit for the ...
1
vote
2answers
51 views

Solve $\frac{1}{2^\theta}\sum_{k=0}^{\theta} {\theta\choose k} \delta(k)=\theta$ for $\delta$

The following arises in unbiased estimation of a parameter for the binomial distribution, but that information is not needed for solving the question. I tried solving this by manipulating the sum to ...
2
votes
4answers
132 views

Prove that $1.49<\sum_{k=1}^{99}\frac{1}{k^2}<1.99$

It can be proven by induction that $$\sum_{k=1}^{n}\frac{1}{k^2}\leq2-\frac{1}{n}$$ From here, we can easily acquire the upper bound of the sum $$\sum_{k=1}^{99}\frac{1}{k^2}$$ letting $n=100$. ...
0
votes
1answer
27 views

Combined pulling power of multiple engines in a train

Apologies if this question doesn't belong here.. Its very common to see multiple engines being used when there are more coaches in a train. In a configuration with more than one engine a good amount ...
0
votes
1answer
15 views

How many ways are there to express a natural as a sum of 3 others—but by induction?

I have figured out an (inductive?) process, but I cannot express it formally: There is always one possibility where $n$ is in the first place of our 3-tuple: $[n~~0~~0]$. Then I can subtract $k~(\leq ...
-4
votes
1answer
47 views

What is the value of this sum? $\sum_{r=k}^{\infty} \frac{1}{r^{3/2} e^{\frac{c k}{2 r}}}$

How can I calculate this, where $c>0$? $$ \sum_{r=k}^{\infty} \frac{e^{\frac{-c k}{2 r}}}{r^{3/2}} $$ One solution might be to upper bound and lower bound this sum by an integral, and then take ...
3
votes
2answers
62 views

How is the Radius of Convergence of a Series determined?

Consider $$\sum_{n=0}^{\infty}\frac{(-1)^nx^n}{(n+1)^2}$$ which by the ratio test the ratio of two consecutive terms converges to $|x|$ as $n\rightarrow \infty$ and has a radius of convergence equal ...
2
votes
4answers
138 views

Summing n times binomial(n,k)

I'm trying to do $\sum_{n=a}^b \left( \begin{array}{rl} n \\ a \end{array} \right) n $ . Is there a formula, that anybody knows?
0
votes
0answers
46 views

Binomial Sum Formula

I can't find a good closed form expression for this, $\sum_{k=0}^n\left[\binom{n}{k}\binom{m}{k}\right]$, where n is the variable, and m is a fixed constant, to be included in the formula. :( Can ...
5
votes
4answers
148 views

Combinatorial Proof for Binomial Identity: $\sum_{k = 0}^n \binom{k}{p} = \binom{n+1}{p+1}$ [duplicate]

I am studying combinatorics and I came across the identity $$\sum\limits_{k=0}^n \binom kp =\binom {n+1}{p+1}.$$ I have read the algebraic proof and it does not appeal to me. Is there an elegant ...
4
votes
0answers
74 views

On finding an explicit form of a particular recurrence relation

Let $f$ be integrable over the interval $[0, 1]$, and $$I_n = \int_0^{1} x^n f(x) \, \mathrm{d}x.$$ Suppose $f(x) = f(1-x)$; we can then show that $$I_n = \sum_{k=0}^{n} \binom{n}{k} (-1)^k \, ...
1
vote
3answers
73 views

Surprising Summation (3): $\sum_{i=1}^n\sum_{j=1}^i 2(n-i)+1=\sum_{i=1}^n i^2$

Show, without expanding the summation, that $$\sum_{i=1}^n\sum_{j=1}^i 2(n-i)+1=\sum_{i=1}^ni^2$$ It would be interesting to see different approaches to this problem, when expansion is not ...