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

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0answers
16 views

Combinatorial identity binomial coefficients [duplicate]

How to prove that $$ \binom{m}{p} = \sum_{j=0}^q \binom{q}{j}\binom{m-q}{p-j}\;?$$
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2answers
15 views

notation for Sumation of Sumation for only for odd iterations

I need to write a summation in summation whether the inner summation should iterate from one through all odd numbers to the teration of the outer summation which goes from 1 to $\infty$... Something ...
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0answers
16 views

Symbol for exponentiation of a sequence? (Equivalent to SIGMA for summation and PI for product)

I have a student asking whether there is a symbol for exponentiation of a sequence? So there's SIGMA for summation of a sequence, PI for multiplication of a sequence and perhaps something else for ...
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1answer
34 views

An expression with gcd and abs is transformed magically!

There's a problem to calculate $\sum^{n}_{i=1}\sum^{m}_{j=1}\frac{|i-j|}{\gcd(i,j)}$, whose tutorial gives the following transformation I really don't understand. ...
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4answers
58 views

How to prove $ \sum\limits_{k=1}^{n}\frac{k}{(k+1)!}=1-\frac{1}{(n+1)!}$ using induction?

This is as far as I get. I get stuck here because both sides to not equal each other, but I am not sure what I am doing wrong. $$ \sum\limits_{k=1}^{n}\frac{k}{(k+1)!}=1-\frac{1}{(n+1)!}$$ Assume: ...
2
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3answers
43 views

Show the closed form of the sum $\sum_{i=0}^{n-1} i x^i$ [duplicate]

Can anybody help me to show that when $x\neq 1$ $$\large \sum_{i=0}^{n-1} i\, x^i = \frac{1-n\, x^{n-1}+(n-1)\,x^n}{(1-x)^2}$$
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1answer
33 views

Need help using ratio test [on hold]

Only using the ratio test determine where the series converges. $$\sum_{n=1}^\infty \frac{8n!}{n^n}$$
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1answer
33 views

Find closed form formula

I need help to find closed form formula for this summation $$\sum_{i=0}^{\infty}(x-y)^i$$
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6answers
923 views

If each term in a sum converges, does the infinite sum converge too?

Let $S(x) = \sum_{n=1}^\infty s_n(x)$ where the real valued terms satisfy $s_n(x) \to s_n$ as $x \to \infty$ for each $n$. Suppose that $S=\sum_{n=1}^\infty s_n< \infty$. Does it follow that ...
3
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0answers
35 views

Help understanding the complexity of my algorithm (summation)

As an exercise, I wrote an algorithm for summing the all elements in an array that are less than i. Given input array A, it produces output array B, whereby B[i] = sum of all elements in A that are ...
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0answers
18 views

It is okay to have a conditions in a summation limit that depend on the current value of another summation

this is one of those things that I know how to do in a programming environment but not sure how it translates into mathematics. I'm trying to express a sum so that it is easily visible that certain ...
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2answers
64 views

How to expand $ \sum\limits_{k=1}^{5}\frac{k}{(k+1)!}$

Can somebody explain step by step how to expand $$ \sum\limits_{n=1}^{5}\frac{k}{(k+1)!}$$ into an equation?
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vote
2answers
47 views

this sum inequality $\sum_{i=1}^{n}\frac{1}{4i(i+1)-1}<\frac{2}{7}$

show that $$\sum_{i=1}^{n}\dfrac{1}{4i(i+1)-1}<\dfrac{2}{7}\tag{1}$$ we have $$4i(i+1)-1>4i^2$$ But $$\sum_{i=1}^{n}\dfrac{1}{i^2}<\dfrac{8}{7}$$ it is clear not hold, because ...
0
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0answers
22 views

Simplifying a formula with summation

I have seen on a data-mining tutorial this sort of formula simplification: to (consider $\bar x$ to be the sum of all $x_i$ dividied by n) So I was wondering if I can apply the same ...
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1answer
45 views

Closed form for the summation [on hold]

Can anyone help me to find what is closed form formula for this summation formula $$\sum\limits_{n=0}^mx^n\sum\limits_{i=0}^ny^i$$
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1answer
48 views

How many unique ways can I sum $k$ non-negative numbers to $N$?

I have a similar question but not exactly the same as this. I'm trying to determine the number of unique multisets $S\in \mathbb{N}$ that exist when the members are required to sum to a number $N$. ...
1
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1answer
27 views

If $ f(n) = \sum_{i = 1}^{n} (n / i) \log(n / i) $ and $ g(n) = n ~ {\log^{2}}(n) $, then is $ O(f) = O(g) $?

I was trying to prove that if $$f(n) = \sum_{i=1}^{n}\frac{n}{i} \log\frac{n}{i} $$ $$g(n) = n \log^2n$$ then $O(f(n)) = O(g(n))$ I am not sure that it is the case, but based on my simulation ...
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0answers
39 views

2015 AMC12A question 25

This is a question from the 2015 AMC12 math competition. I haven't really made much progress at all on it, and I just want to know the right way to solve this equation. A collection of circles in ...
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2answers
26 views

solving the limit $\lim_{n\to \infty}\sum_{k=1}^n|e^{(2πik)/n}-e^{(2πi(k-1))/n}$|

$$\lim_{n\to \infty}\sum_{k=1}^n\left|e^{(2πik)/n}-e^{(2πi(k-1))/n}\right|$$ i can solve it geometrically. but is there any way to solve it using Euler's formula ?, the answer will be one of these ...
3
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3answers
70 views

Does $\sum_{n=1}^\infty \frac{\cos(n\pi/3)}{n!}$ absolutely converge?

Using the Ratio Test, I have to find whether $$ \sum_{n=1}^\infty \frac{\cos(n\pi/3)}{n!} $$ converges or diverges. The back of the book says that the sum is absolutely convergent. My work: $a_n = ...
3
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2answers
63 views

Show that if $a\neq 1$, then $\sum_{k=0}^{n-1}ka^k = \frac{1-na^{n-1}+(n-1)a^n}{(1-a)^2}$

Need to show that if $a\neq 1$, then $$\sum_{k=0}^{n-1}ka^k = \frac{1-na^{n-1}+(n-1)a^n}{(1-a)^2}$$ Here is my attempt: $$\begin{aligned} S & =\sum_{k=0}^{n-1}ka^k \\ &= ...
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1answer
37 views

Find the closed-form for $\sum_{i=0}^n(-1)^i(\frac{1}{2})^i$

I start with simplifying: $$\sum_{i=0}^n(-1)^i(\frac{1}{2})^i=\sum_{i=0}^n(-\frac{1}{2})^i$$ then: $$S = 1 + (-\frac{1}{2}) + (-\frac{1}{2})^2 + ... +(-\frac{1}{2})^n$$ $$(-\frac{1}{2})S = ...
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1answer
15 views

What is the sum over a shifted sinc function?

What is the sum of a shifted sinc function: $$g(y) \equiv \sum_{n=-\infty}^\infty \frac{\sin(\pi(n - y))}{\pi(n-y)} \, ?$$
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4answers
90 views

Prove $\sum_{i=2}^{n}\frac{1}{(n-1)n}$ = $\frac{(n-1)}{n}$ using induction.

I need to prove $\sum_{i=2}^{n}\frac{1}{(i-1)i}$ = $\frac{(n-1)}{n}$ using induction. I am getting stuck midway through the inductive step. Here is what I have: $\forall n\geq 2$, where ...
3
votes
4answers
77 views

Sum of Square roots formula.

I would like to know if there is formula to calculate sum of series of square roots $\sqrt{1} + \sqrt{2}+\dotsb+ \sqrt{n}$ like the one for the series $1 + 2 +\ldots+ n = \frac{n(n+1)}{2}$. Thanks in ...
0
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4answers
52 views

Solution verification: $\sum_{n=1}^\infty \frac{9^n}{3+10^n}$

I need to find out whether $$\sum_{n=1}^\infty \frac{9^n}{3+10^n}$$ converges or diverges using the limit comparison test. Here's my work: Let $a_n$ be $\frac{9^n}{3+10^n}$, $b_n$ be ...
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1answer
81 views

Do we have $\sum_{n=1}^\infty 0=0$?

Simple question: Do we have $$\sum_{n=1}^\infty 0=0$$ ? Mathematically this seems obvious, but in practice I am very uncomfortable with this. Because nothing is perfect, so $0$ might not be quite ...
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1answer
12 views

$\phi$, and the uses of an alternate formula

I was trying to find the solution to the formula: $$x = \sum_{n=1}^\infty{x^{-n}}$$ I found it to be the golden ratio, or $\phi = \frac{1 + \sqrt{5}}{2}$. I do not know if this has already been ...
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1answer
48 views

Gauss Method to show [on hold]

Could you please give me the way to solve this problem Using Gauss method to show if $x ≠ y + 1$ then $$ \sum_{i=0}^n (x-y)^i = \frac{(x-y)^{n+1}-1}{x-y-1}. $$
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1answer
27 views

Determine the radius of convergence of $\sum_{n=1}^\infty n^{n^{1/3}}z^n$ (by the ratio test if possible)

Determine the radius of convergence of the following power series: $\sum_{n=1}^\infty n^{n^{1/3}}z^n$ Applying the ratio test gives $\frac{({n+1})^{({n+1})^{1/3}}}{n^{n^{1/3}}}z<1$. So ...
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0answers
15 views

Prove by induction: $E[\sum_{i=1}^nc_iU_i(X)]=\sum_{i=1}^nc_iE[U_i(X)]$ Please just check what I've done

Prove by induction: $$E[\sum_{i=1}^nc_iU_i(X)]=\sum_{i=1}^nc_iE[U_i(X)]$$ Let me show you what I've done. I think I'm right: $$n=1,$$ $$E[c_1U_1(X)] = c_1E[U_1(X)]$$ Okay so maybe this one looks ...
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2answers
56 views

Value of the sum: $\binom{19}{0} - 1/2\binom{19}{1} +1/3\binom{19}{2} - 1/4\binom{19}{3} … -1/20\binom{19}{19}$?

How do I find the value of this sum? I tried taking out the equal binomial coefficients as factors but this didn't really simplify anything. I am stumped.
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2answers
53 views

How we can prove that: $\sum _{k=n}^n f\left(\frac{k}{n}\right)\le n\cdot \log(2)$?

$f:\left[0,1\right]\rightarrow R,\:f(x)=\frac{1}{1+x}$ and we have to show that $\sum_{k=n}^n f\left(\frac{k}{n}\right)\le n\cdot\log(2)$. What I know is just that: $n\cdot \log(2)=\int_0^1 ...
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1answer
56 views

How simplify this sum?

I need help to simplify this sum : $$\sum_{i=0}^{x-1}\left(1-\dfrac{1}{2^i}\right)^{m-1}$$ Is it possible to simplify it ? Thank you.
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0answers
55 views

When can we assume this identity about series?

Suppose we are given that $$\sum_{i =a} ^b f (i)g (i) =\sum_{i =a} ^b f (i)h (i) \tag {1} $$ when can we conclude that $$\forall i =a,…,b:g(i)=h(i) \tag {2} $$ ? For example, if $g, h $ are both ...
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2answers
85 views

Find the sum of series $(1^2+1)1!+(2^2+1)2!+(3^2+1)3!+…+(n^2+1)n!$.

Find the sum of series $(1^2+1)1!+(2^2+1)2!+(3^2+1)3!+...+(n^2+1)n!$ I have found one method as i have shown in my answer below. But that form took me 30 mins to identify. ...
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1answer
66 views

evaluate trigonometric

To all the genius out there , here is a question about expresssing summation of hyperboilc functions : First of all, I've already proved that: $$\sinh(x + 1)- \sinh(x) = (-􀀀1 + \cosh (1)) \sinh(x) ...
3
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1answer
61 views

Is there a formula for the closed form for $ \displaystyle \sum_{r=1}^\infty \frac{\sum_{k=1}^r k^n}{r!}$ for any positive integer $n$?

Is there a formula for the closed form for $ \displaystyle \sum_{r=1}^\infty \frac{\sum_{k=1}^r k^n}{r!}$ for any positive integer $n$? I tried Faulhaber's formula and Bell number but couldn't ...
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4answers
133 views

How to prove that the value of $e$ is irrational without using the number $e$ itself [on hold]

Recently I have tried to prove that the value of $e$ is irrational without using the number $e$ itself. I have seen that the number $e$ can be expressed as $$\lim_{n\to\infty}(1 + 1/n)^n;$$ however, ...
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1answer
39 views

Sum of absolute values is finite

Suppose $\lambda_m \in \mathbb{R}$ and suppose that $\sum_{m \in \mathbb{N}} \lvert i + \lambda_m \rvert^{-p} < \infty$. Then why does $$\sum_{(m,n) \in\mathbb{N}\times\mathbb{Z}} \left\lvert i \pm ...
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0answers
23 views

Estimating a sum

i want to show the following: Assume that $\sum_{m\in\mathbb{N}}{|i+\lambda_m|^{-p}}<\infty$ (where $(\lambda_m)_m$ is a sequence of real numbers). I want to show that then also holds for each ...
3
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1answer
48 views

How we can show that $\:I_n\ge \frac{2}{\pi }\left(\frac{1}{n+1}+\frac{1}{n+2}+…+\frac{1}{2n}\right)$

We have $I_n=\int _{\pi }^{2\pi }\:\frac{\left|sin\left(nx\right)\right|}{x}\:dx,$ and we need to show that$\:I_n\ge \frac{2}{\pi }\left(\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{2n}\right)$ I write ...
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vote
2answers
53 views

radius of convergence of $\sum_{n=1}^\infty n!^2x^{n^2}$ [closed]

Determine the radius of convergence of the following power series: $\sum_{n=1}^\infty n!^2x^{n^2}$
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1answer
43 views

Infinite summation of a trigonometric series

$\sum_{n=1}^{n=\infty}\sin(\frac{n\pi x}{L})\sin(\frac{n\pi y}{L})\surd(k^2+\frac{n^2 \pi^2}{L^2})$ I am trying to solve the above summation. I still could not figure out if this summation converges ...
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1answer
62 views

Simplifying Sum

How would one show that $$ \sum_{i=0}^n\binom{n}{i}(-1)^i\frac{1}{m+i+1}=\frac{n!m!}{(n+m+1)!} ? $$ Any hint would be appreciated. Note: I tried to recognize some known formula, but since I don't ...
0
votes
0answers
19 views

Help with simplification rules form sums and integrals.

IF you had a power series with summation notation and an integral what expressions would you be able to pull outside the integral and which would you be able to pull outside the sum.
3
votes
1answer
413 views

Simplification of a double sum involving partial sums of harmonic series

Could somebody explain the jump in the following equation? $$\frac{1}{n}\sum\limits_{i=1}^{n}\left[1 + \sum\limits_{j=i+1}^{n}\frac{1}{m}\right] = 1 + \frac{1}{nm}\sum\limits_{i=1}^{n}(n-i) $$
0
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0answers
33 views

Sum of combinatorics

I encounter the following sum of combinatorics and I can not seem to find the solution. Can anyone lend a hand? $\binom {n^c} {a} \binom {n^{c'}}{2a} + \binom {n^c} {a+1} \binom ...
2
votes
5answers
92 views

How to sum $\sum_{k=1}^n (k+1)(k)(k-1)$

Is there an intelligent way to do this sum without using sums of cubes and sums of squares? $$\sum_{k=1}^n (k+1)(k)(k-1)$$
1
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2answers
35 views

Taking the square of a finite series

I was reading something that involved taking the square of a sum $\sum_{i=1}^k(n_i-1)$. The author just presented the result. $$\left(\sum_{i=1}^k(n_i-1)\right)^2 = \sum_{i=1}^k(n_i^2-2n_i)+k + cross ...