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

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0
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2answers
24 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
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2answers
192 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
83 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
50 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
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1answer
26 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{ ...
2
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0answers
48 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
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2answers
50 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 ...
1
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2answers
54 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
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0answers
52 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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0answers
29 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
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3answers
32 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 $$\forall n \geqslant 2: \sum_{i=1}^{n} \frac{4}{5^{i}} < 1$$
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5answers
93 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
48 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
2answers
82 views

Show $\lim_{x \to0^+} \sum_{n=1}^{\infty} \frac{2x}{n^2x^2+1} = \pi$

Show that: $$\lim_{x \to0^+} \sum_{n=1}^{\infty} \frac{2x}{n^2x^2+1} = \pi$$
1
vote
3answers
65 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
180 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
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1answer
37 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
21 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
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1answer
88 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
41 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} ...
17
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3answers
407 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 ...
0
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0answers
43 views

Simple proof explanation - Possibly triangle inequality involved

I'd like some help with understanding the following statements...I saw it on the internet while searching for a proof, and I'd like to understand why its true: let $A$ be a diagonally dominant matrix ...
1
vote
1answer
45 views

Can this double sum be simplified?

Can this summation be simplified ($A$ is a constant)? $$\displaystyle\sum_{i=1}^{A} \displaystyle\sum_{j>i}^{A} f(i)f(j)$$ By simplified I mean either a closed-form expression in terms of $A$ ...
0
votes
1answer
47 views

$\sum (-1)^{n+1} n^{1/n} $ is not converging?

why is $\sum (-1)^{n+1} n^{1/n}$ is not converging? What is meant by convergence here, is the value does not stay firm?
2
votes
0answers
58 views

How to prove these indentities? [closed]

How to prove these indentities? $\displaystyle \sum \limits_{k\geq0} {2n\choose 2k-1}{k-1\choose m-1}=2^{2n-2m+1}{2n-m\choose m-1}$ $\displaystyle \sum \limits_{k=0}^{m} {m\choose k}{n+k\choose ...
2
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1answer
50 views

Calculate sum wtih binomial coefficients

I need help with finding the sum of $\sum \limits_{k=0}^{n} \frac{1}{k+1}{n\choose k}x^{k+1}$
0
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2answers
45 views

How to calculate this sum

How do you calculate this sum $ \sum \limits_{k=1}^{n} \frac{k}{n^k}{n\choose k}$ ?
2
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2answers
26 views

School Play and Ticket problem.

I may be missing something obvious here, but cant seem to see what. Can anyone give me some insight on how to solve this. 100 tickets are sold for a school play. Tickets for a child cost £1.50 each. ...
0
votes
2answers
47 views

Answer clearly wrong; what went wrong?

I was trying to answer this question Not sure which test to use? Here was my clearly wrong response: $\ln(1 + \frac{1}{3^k})$ = $\frac{\ln((1+\frac{1}{3k})^k)}{k}$ $\sum_{k=1}^\infty ...
2
votes
1answer
86 views

How to numerically evaluate the CDF of this random variable?

I have a discrete random variable $X=0,1,2,\ldots$ with the following probability mass function: $$P(X=x)=\sum_{t=0}^x\binom{n}{t}p^t(1-p)^{n-t}\binom{m-t}{x-t}q^{x-t}(1-q)^{m-x}\tag{1}$$ where ...
0
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0answers
57 views

Nested sums and products

I am trying to determine how to express this series in a general form as a summation of products: \begin{equation} \begin{aligned} (p_{i1}p_{j1})(1-p_{i2}p_{j2})(1-p_{i3}p_{j3}) \mbox{ } &+ \\ ...
0
votes
1answer
39 views

Find beginning of the asymptotic expansion of the sum

Find beginning of the asymptotic expansion of the sum: $$ (n!)^{-1}\sum^{n}_{k=1}k! $$ against the function $n^{-i}$, for $i\geq0$ to the nearest $\mathcal{O}(n^{-5})$.
2
votes
4answers
102 views

Exact value of $\sum\limits_{n=1}^\infty(-1)^{n(n+1)/2}/n$?

Wolfram is not computing it properly. What is the exact value of $$\sum_{n=1}^\infty\frac{(-1)^{n(n+1)/2}}{n}?$$ How to avoid imaginary $i$ coming from the exponent?
2
votes
2answers
38 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 ...
3
votes
2answers
60 views

Help finishing proof via induction for a summation

So I have to prove the following equation using induction for n >= 2: $$ \sum\limits_{i=1}^n 4/5^i < 1 $$ However the question asks me to prove something stronger such as this: $$ ...
5
votes
1answer
72 views

Question about $ \int_{-1}^{0}\sum_{n=1}^{x}n^sdx=\zeta (-s) \forall s\in \Bbb N$

what I found from messing around was $$ \int_{-1}^{0}\sum_{n=1}^{x}n^sdx=\zeta (-s) $$ $$ s\in \mathbb{N} $$ when the partial sum is changed to an equivalent polynomial using Faulhaber's formula. ...
3
votes
2answers
110 views

Evaluate the summation involving binomials.

$\sum _{ i=0 }^{ 100 }{\binom{k}{i}}*{\binom{M-k}{100-i}*\frac{k-i}{M-100}}/{\binom{M}{100}}$ I wrote the first few terms but couldn't find any pattern and how to club the terms. Help.
2
votes
1answer
85 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
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3answers
46 views

Evaluating $\sum_{k=1}^{30} k(30-k)$

I tried to rewrite it as $\sum_{k=1}^{30} k(30-\sum_{k=1}^{30}k)$ and then replace the $\sum_{k=1}^{30} k$ with $\frac{n(n+1)}{2}$ then substitute $n=30$ into the equation, however I am not getting ...
1
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2answers
28 views

Difference between $\lim_{n\rightarrow\infty}\sum_{k=1}^{n}a_k$ and $\sum_{k=1}^{\infty}a_k$

Is there any difference between $\lim_{n\rightarrow\infty}\sum_{k=1}^{n}a_k$ and $\sum_{k=1}^{\infty}a_k$? My example and thought: Let $a_n=n$ where $n\in\mathbb{P}$. $\mathbb{P}$ is the set of all ...
3
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1answer
74 views

Is there some way to simplify $\sum_{i=1}^n \sum_{j\neq i}(\frac{j-1}{2})(\frac{i-1}{2}) $ To obtain a closed form.

Is there some way to simplify $\sum_{i=1}^n \sum_{j\neq i}(\frac{j-1}{2})(\frac{i-1}{2}) $? Does it have a closed form? It's the last piece of a puzzle I need to solve a similar question ...
3
votes
2answers
80 views

A summation identity which is for me hard to verify

I found a summation identity when I try to figure out the proof of an exercise in Apostol's Mathematical Analysis (Page 27, Exercise 1.26): $\ ~~~~$ If $a_1\geq a_2\geq \dots\geq a_n$ and $b_1\geq ...
2
votes
2answers
70 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} ...
0
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3answers
76 views

Can we pull $f$ from this equation?

$$\sum_{k=1}^\infty f(k)\cdot e^{-k} = c$$ Can we pull $f$ function from this equation somehow? $c$ is a real constant...
0
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2answers
20 views

Lowering the start indexes in sums

I'm implementing a CYK algorithm in my software and I've found a pseudo-code on Wikipedia. Here's its complexity(modified version for special use, which doesn't go from ...
4
votes
2answers
98 views

Finding the summation of the floor of the series identity

I would appreciate if somebody could help me with the following problem: Q: How to proof ? The number of positive divisors of $n$ is denoted by $d(n)$ ...
0
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2answers
54 views

Summation of$ n^3\over (3n)!$

Summation of $n^3\over(3n)!$ from 1 to infinity Please provide procedure. Don't know how to proceed. Tried making it to form $n^2\over3 (3n-1)!$ then?
3
votes
1answer
76 views

Sum of combinations series

What is the value or tight upper limit of the following summation: $$\sum_{k=0}^n{n\choose k} x^{k(n-k)}$$
3
votes
1answer
60 views

How find this sum $\sum_{n=0}^{\infty}(-1)^{n}\frac{n+1}{(2n+1)!}$

Find this follow sum $$\sum_{n=0}^{\infty}(-1)^{n}\dfrac{n+1}{(2n+1)!}$$ My try:since ...
2
votes
2answers
125 views

Solve $2xy'' + 5y' + xy = 0$ using Frobenius method?

Using the Frobenius method, solve the differential equation $2xy'' + 5y' + xy = 0$. I've done most of the work, but when it comes to getting the indicial equation I am getting stuck. When working ...