Questions tagged [summation]

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

0
votes
4answers
60 views

Summation $k=n-4$ to $k=4$ of $1$

This is perhaps very basic but I am currently very lost on how to think in order to end up with the answer: $9-n$ in the following summation: $$\sum_{i=n-4}^4 1 = 9-n $$ My first idea was to ...
0
votes
4answers
114 views

Calculate the following sum

Calculate the following sum $$\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n(n+1)}$$ I Tried decomposing the denominator but still seem to be stuck thanks to the numerator
12
votes
3answers
239 views

Basel Problem - Area of $\frac 16$ of Circle with Radius $\sqrt{\pi}$.

There are several proofs to the solution of the well-known Basel Problem, i.e. $$\sum_{n=1}^\infty \frac 1{n^2}=\frac {\pi^2}6$$ Is is possible to create a geometrical interpretation of this ...
7
votes
3answers
518 views

Deriving sum of powers formula using generating functions

Just for fun I wanted to try to derive a formula for the sum of $p$-powers using generating functions, but without using any literature or websites for help. However I do need a small push or hint. ...
4
votes
3answers
81 views

Interpretation of sums using $\cdots$

Consider the sum $$\sum_{1\le k_1 < k_2 < \cdots < k_r \le n}k_1k_2\ldots k_r$$ Does this simply mean $$\sum_{\substack{|K|=r}\\\inf(K)\ge 1\\\sup(K)\le n}\prod_{k\in K} k$$ I am ...
4
votes
3answers
61 views

Estimating the Lower Bound of A Summation Related to Probability

I am working on a probability problem which requires me to find a lower bound of a sum. The sum is $$\sum_{i=n}^{100}{100\choose i}\left(\frac{80}{100}\right)^i\left(\frac{20}{100}\right)^{100-i}\geq ...
3
votes
3answers
212 views

Number equal to the sum of digits + product of digits)

Are every number equal to (sum of digits + product of digits) in a given base only two digits long ? Thought about limiting like this : $$b^{(n - 1)} \leq N = \text{Product digits} + \text{Sum ...
3
votes
3answers
119 views

Trying to understand polynomials proof of Vandermonde's identity.

In Spivak's calculus, page 28, problem No.4, Asked to prove : $$\sum_{k=0}^l {n \choose k}{m \choose l-k}={n+m \choose l}$$ by using $(1+x)^n(1+x)^m=(1+x)^{n+m}$ The answer book comes quickly from : $...
1
vote
3answers
50 views

How do I go about solving various summation of binomial coefficients like $\sum_{r=0}^{n} \binom{n}{r}f(r)$

I've come across many problems that require me to find summation of binomial coefficients. How go about solving these kind of summations of the form $$\sum_{r=0}^{n} \binom{n}{r}f(r)$$ where f(r) ...
1
vote
3answers
216 views

What is the $m^{th}$ derivative of $-e^{-\alpha\sum_{k=0}^Kv_kx^k}$

$$\huge-e^{-\alpha\sum_{k=0}^Kv_kx^k}$$ My attempt: $$\large(-e^{-\alpha\sum_{k=0}^Kv_kx^k})^{(m)}=(-e^{-\alpha\sum_{k=0}^Kv_kx^k})(\sum_{k=1}^Kv_kk(k-1)\cdots(k-m+1)x^{k-m})$$ Is this correct? ...
1
vote
3answers
122 views

Find the SUM of all the numbers of S.

Let S be the set of all three digits numbers. Such that The digits in each number are all from the set $\left\{1,2,3,\ldots,9\right\}$. Exactly one digit in each number is even. Then, find the SUM ...
1
vote
3answers
1k views

How do I prove that the sum: $1/\ln(n)^p$ diverges for $p>1$

So I need to prove that the infinite sum $\frac{1}{(\ln(n)^p)}$ diverges for all values of $p$. I managed to prove it for $p\leq 1$ via comparison test with $1/n$. but for this I can't seem to find a ...
1
vote
3answers
70 views

How to prove $\sum_{n=1}^{\infty} \frac{3^n +7n}{2^n (n^2+1)} $ diverges?

$$\sum_{n=1}^{\infty} \frac{3^n +7n}{2^n (n^2+1)} $$ It seems clear to me that this seires diverges since the dominant term is $(3/2)^n$, a geometric series with $r > 1$ However I am required to ...
1
vote
3answers
305 views

Can I approximate a series as an integral to find its limit and determine convergence?

Find $\lim \limits_{n \to \infty} (a_n)$, where $a_n=\frac{1}{n^2}+\frac{2}{n^2}+\frac{3}{n^2}+...+\frac{n}{n^2}$. So I can solve it like that $a_n=\frac{1+2+3+...n}{n^2}=\frac{\frac{1}{2}n(n+1)}{n^2}...
1
vote
3answers
929 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
3answers
46 views

Why is $\sum\limits_{i = 1}^n \frac{n}{n-i+1}$ equal to $n \sum\limits_{i = 1}^n \frac{1}{i}$?

Assume we have the sum $\sum\limits_{i = 1}^n \frac{n}{n-i+1}$ why is this equal to $n \sum\limits_{i = 1}^n \frac{1}{i}$?
0
votes
3answers
43 views

How do you compute ab+i in terms of a and b

Given that $a+bi=p$ and $a^2-b^2=q$, how can I compute $ab+i$? Note $i=\sqrt{-1}$ is the complex number $a$ and $b$ can be any integer
0
votes
3answers
88 views

Solving a recurrence relation: can't figure out how to convert from summation

I am really struggling to solve this recurrence. $$ T(n) = T(\sqrt{n}) + n. $$ I am asked to give asymptotic upper and lower bounds for $T(n)$. I am free to use any method to arrive at my answer, ...
0
votes
3answers
37 views

Calculate $\sum\limits_{m=0}^{\infty}\sum\limits_{i=0}^{m}\frac{1}{m!} \binom{m}{i} a^{i}b^{m-i} $

Prove that $$\sum\limits_{m=0}^{\infty}\sum_{i=0}^{m}\frac{1}{m!} \binom{m}{i} a^{i}b^{m-i}. =\sum\limits_{i=0}^{\infty}\sum_{m=i}^{\infty}\frac{1}{i!(m-i)!}a^{i}b^{m-i} $$ I'm stacked how changed ...
0
votes
3answers
58 views

Binomial summation with alternating terms?

How do I solve problems of type :$$\sum_{k=1}^{(n+1)/2}\binom n{2k-1}x^k\text{ or }\sum_{k=0}^{n/2}\binom n{2k}x^k$$ I tried transferring the binomial to $n-1$ but the repeating $x^k$ makes it weird. ...
0
votes
3answers
71 views

A question proving a variant of the handshake theorem

Let $n \in \mathbb{N}$, and assume $n≥1$. Suppose you are at a party with $n$ people (including yourself). At the end of the party, define a person’s parity as odd if they have shaken hands with an ...
0
votes
3answers
52 views

Why are these summations not equivalent?

Hello i have this sum: $\sum_{j=0}^{n-2}(n-j)$ i try solve of this mode: $\sum_{j=0}^{n-2}n - \sum_{j=0}^{n-2}j$ or $\sum_{j=0}^{n-2}n + \sum_{j=0}^{n-2}-j$ but in wolfram alpha it not same, and ...
0
votes
3answers
95 views

How to explain this summation reordering?

Could anyone help me explain how can we take the $x$ out of the sum and reorder the summation? Let $\Omega$ be countable. Then, every random variable $X:\Omega\to\mathbb{R}$ is discrete, and ...
-1
votes
3answers
20 views

Solve $\sum_{i=1}^n n*x^{2n}$

I know that $\sum_{i=0}^n n*x^n = \frac{x}{(1-x)^2}$ but the factor of 2 in the exponential makes it a riemann zeta function? Can anyone shed some light on this?
-1
votes
3answers
49 views

How can I calculate the sum of $\sum\limits_{n = 1}^{\infty} n(n+3)x^n$?

How can I calculate the sum of $\sum\limits_{n = 1}^{\infty} n(n+3)x^n$ analytically?
0
votes
2answers
374 views

Sums of Geometric Series Induction

Suppose that today you put 1000 dollars in a deposit account that pays you 1% interest every month. After N months, the account balance will be $ 1000 \times 1.01^N $ dollars. Now consider a variant ...
0
votes
2answers
63 views

$\sum_{k=0}^{n} a_{k}=\sum_{k=0}^{\lfloor n/2 \rfloor} a_{2k} +\sum_{k=0}^{\lfloor (n-1)/2 \rfloor}a_{2k+1}$

I would like to show that \begin{align} \sum_{k=0}^{n} a_{k}&=\sum_{k=0}^{\lfloor n/2 \rfloor} a_{2k} + \sum_{k=0}^{\lfloor (n-1)/2 \rfloor}a_{2k+1}\\ \end{align} I'm interested in ...
0
votes
2answers
110 views

Solving for a variable in a summation problem?

$$x=\sum_{N=1}^TA^N$$Say I have a problem like this , how would go about rewriting this equation so that I can solve for t using x and a? I don't know what the individual variables are called so I ...
0
votes
2answers
80 views

Proving that a summation is multiplicative

I have been give a project for number theory: For $m>0$ , let $f(m) = \sum_{r=1}^m \frac{m}{\gcd(m,r)}$ . Evaluate $f(m)$ in terms of the prime factorization of $m$. So far, I have found a formula ...
0
votes
2answers
42 views

Understanding an expression involving sum notation

See the following expression. $$\sum_{i=1}^n \sum_{j \in i} f(a_i)d_j - \sum_{i=1}^n \sum_{j \in i} f(b_j) d_j$$ So in the first two sums, we pick a $i$, and then we sum over all $j$ that satisfy $j \...
0
votes
2answers
39 views

Discrete Math-Computing Summations

So I'm asked to compute a summation with an upper limit $k = 20$ and lower limit $k=1$, where: $B_k= 0$ when $k=1$, and $B_k = \dfrac{1}{(k^2-1)}$ , for $k>1$. I was wondering if there is a ...
0
votes
2answers
71 views

Long summation question, including sets

I have a really long question I'm absolutely stuck on, I don't even know where to begin: Given: $n \in \mathbb{Z}, \geq 2$ let $S$ be the set of all nonempty subsets of {2,3,...,n}. For each $S_i \...
0
votes
2answers
49 views

How to take this derivative

My question is straightforward: I need to evaluate an expression of the form $$ \frac{\partial}{\partial t}\sum_{k=0}^{t}\varphi(k,t) $$ How is this done, usually?
0
votes
2answers
140 views

summation of polynomial products

I need help in understanding how the summation of the product of two polynomials is written. $(a_{0} +a_{1}x +a_{2}x^{2})(b_{0} +b_{1}x + b_{2}x^{2}) =\\ (a_{0}b_{0})x^{0} + (a_{1}b_{0} + b_{1}a_{0})...
0
votes
2answers
44 views

Average length of a bitstring

I am trying to compute the average length of a bit string from all bit strings of $\{0,1\}^n$. By length n I mean a bit string of length n where the most significant bit is 1. I know there is $2^0$ ...
0
votes
2answers
129 views

Compound interest with a compounding interest rate

I have an investment which pays 3% interest (r) annually but it also increases the interest rate every year by 5% (g). I re-invest all interest payments at the start of each year. How many years (t) ...
0
votes
2answers
113 views

Samplification of a sum of multiplication

Supposing I have the following sequence based on two indexes: $a$ and $b$. For $a$ starting with $1$ and $b$ starting with $5$ we have the following sum: $$1 \cdot 5 + 2 \cdot 4 + 3 \cdot 3 + 4 \...
0
votes
2answers
49 views

How should I solve this summation problem?

Lets say that we have these $x$ and $y$ coordinates $x=1,2,3,4,5$ and $y=6,7,8,9,10$ and where $n=5$. How would I use these $x$ coordinates with the first summation? Now, I know that learning is ...
0
votes
2answers
52 views

Inequation of an sum smaller than 1

I'm trying to figure out the following $$ \sum^{\infty}_{n=3} \dfrac{q!^2}{n!^2} < 1 $$ How I can show it if $q \geq 2$? Maybe with telescoping sums? Thanks, Landau
0
votes
2answers
48 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⌉ OP has ...
0
votes
2answers
186 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
144 views

finite sum over a Gaussian

I have a sum of the form: $$\sum_{n,m=-N}^N e^{-\alpha (n-m)^2}$$ where $\alpha > 0$ is some constant, and $n,m$ take the integer values: $-N,..,N$. I know there is a possibility of exchanging ...
0
votes
2answers
963 views

Calculating a Summation Involving 2 Variables

Having not taken a math course for multiple years, I appear to have forgotten some bare basics. Unfortunately, Google has not taken me to a solid answer after much searching. How do you solve an ...
0
votes
2answers
124 views

Simple addition of summation

I know the following equality holds from previous work $\sum_{i=1}^{n}a_i^2\sum_{j=1}^{n}b_j^2 + \sum_{i=1}^{n}b_i^2 \sum_{j=1}^{n}a_j^2$ = $2(\sum_{i=1}^{n}a_i^2)( \sum_{i=1}^{n}b_i^2)$ But when I ...
-1
votes
2answers
45 views

What is the difference between the summations?

What is the difference between the summation $$\sum_{1 \leq i<j \leq n} f(i,j)$$ and $$\sum_{1\leq i} \sum_{<j \leq n} f(i,j)?$$
-1
votes
2answers
244 views

Calculating double sum

How do I calculate this sum (as a function of n)? I have no experience in calculating sums like these, so I don't know any of the rules regarding this subject.
-1
votes
2answers
48 views

Multiplication problem

I have a very simple question As 0.999⋯≠1 Hence, ...
-1
votes
2answers
172 views

Finding a closed form solution

For the sequence 0 2 8 34 144 ... The recurrence relation is: $$\begin{align*} E(n) = 4*E(n-1)+E(n-2) \end{align*}$$ How to calculate the closed form expression ...
-1
votes
2answers
78 views

Does limit exist for the following expression?

If limit exists, then what is its value? And if it does not exist then can we find where does this expression tends as $ n \to \infty$. The expression : $\lim\limits_{n \to \infty } \sum\limits_{k=...
-2
votes
2answers
38 views

Can anyone explain how this summation was simplified/rearranged? I can't for the life of me follow the steps

See attached image. Part of a question asking to calculate the marginal pmf when the joint pmf is known.