Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [summation]

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

The notation $\sum\limits_{i=1}^na_i$ means $a_1+\ldots +a_n$.

Use for sums of infinite series and questions of convergence; use for questions about finite sums and simplification of expressions involving sums.

17770 questions
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Showing $\left\lfloor \sum_{k=1}^{10000} {1 \over \sqrt k}\right\rfloor = 198$

Some time ago, I was given a homework, which, among other things, asked to demonstrate the following equality: $$\left\lfloor \sum_{k=1}^{10000} {1 \over \sqrt k}\right\rfloor = 198$$ I've tried a few things but nothing got me too far; I haven't…
Eckchart
  • 63
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$\sum_{n=0}^{\infty} \sum_{k=0}^{\infty} \frac{n}{(k+n)!} \frac{n}{k!}$

$\displaystyle{\sum_{n=0}^{\infty} \sum_{k=0}^{\infty} \frac{n}{(k+n)!} \frac{n}{k!}} ~= ~?$ From numerical calculation, I believe the answer is $e^2$. However, I have no idea how to show this.
rudgns55
  • 624
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5 answers

Double Sum with a Neat Result

Based on an interesting question here (second question), I have devised a similar one. Evaluate the following double sum without expansion and substitution of standard sum-of-integers formula. $$\sum_{x=1}^n\sum_{y=1}^n (n-x+y)$$
Hypergeometricx
  • 22,657
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How to evaluate $\sum\limits_{k=0}^{n}\arctan f(k)$ where $f(k)$ is a rational fraction

Find the sum closed form $$\sum_{k=0}^{n}\arctan{\dfrac{k^4+6k^3+10k^2-k-9}{(k+1)(k+2)(k+3)(k^3+7k^2+15k+8)}}$$ For problems involving sums, the idea is to use trigonometricidentities to write the sum in the…
math110
  • 93,304
6
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10 answers

Find the sum of the squares of the first $n$ natural numbers

I've been asked to find the sum of the squares of the first $n$ natural numbers. My initial thought was to just program a brute-force solution but I was wondering if there is a mathematical formula to come up with the answers. I've made a table of…
user343576
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5 answers

Do summations always increment by one? How do you incrment by a negative number? Or any real number?

Do summations always increment by one? Having a more of background in programming than math. I am just learning about summations and I look at them as loops that increment by one. If my assumption correct what an equivalent method / form to…
user1787331
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How to evaluate $\sum_{n=1}^m 2^n \arctan 2^n \theta$

I need to evaluate $$\sum_{n=1}^m 2^n \arctan 2^n \theta$$ as a function of $m$ and $\theta$. All I've done so far is write out the series explicitly: $$\sum_{n=1}^m 2^n \arctan 2^n \theta = 2 \arctan 2\theta + 4\arctan 4\theta + 8\arctan 8\theta +…
Zain Patel
  • 16,802
6
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The sum of squares of the first $n$ natural numbers.

My basic question is this: how to find the sum of squares of the first $n$ natural numbers? My thoughts have led me to an interesting theorem: Faulhaber's formula. It is known that $$1^k+2^k+\ldots+n^k=P_{k+1}(n)$$ is a polynomial of degree $n$…
PersonaNonGrata
  • 955
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1 answer

sum of center binomial coefficients over exponential

I'm trying to find a closed form for the following sum, if anyone knows a way, a hint would be much appreciated... $$ X(n) = \sum_{i=1}^n \frac{i \choose \left \lfloor{i/2}\right \rfloor }{2^i}\ $$ The first few terms are: $$ 1 : 0.5\\ 2 : 1.0 \\ 3…
qwerty314
  • 756
5
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4 answers

What's $\sum_{k=0}^n\binom{n}{2k}$?

How do you calculate $\displaystyle \sum_{k=0}^n\binom{n}{2k}$? And doesn't the sum terminate when 2k exceeds n, so the upper bound should be less than n? EDIT: I don't understand the negging. Have I violated a rule of conduct or something?
Caddyshack
  • 258
5
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6 answers

Calculate $\sum_{k=1}^n \frac 1 {(k+1)(k+2)}$

I have homework questions to calculate infinity sum, and when I write it into wolfram, it knows to calculate partial sum... So... How can I calculate this: $$\sum_{k=1}^n \frac 1 {(k+1)(k+2)}$$
MathDav
  • 556
5
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1 answer

Calculate the sum of S.

Consider $n\in\mathbb{N}.$ Find the sum of:$$S=\left(\dfrac{C_n^0}{1} \right)^2+\left(\dfrac{C_n^1}{2} \right)^2+\cdots+\left( \dfrac{C_n^n}{n+1}\right)^2$$ I don't know how to solve it, i don't have any ideas, so i am very happy to hear from you.…
I Love Math
  • 213
5
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1 answer

Binomial summation without coefficient

I have the following summation: $$\sum_{i=0}^na^{n-i}b^i$$ I recognise that if the binomial coefficient was present, it would represent the expansion of $(a+b)^n$. However, since that coefficient is absent, I am struggling to find a formulaic…
eggyal
  • 195
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2 answers

How show that this sum $\sum^{n}_{i=0} \frac{(-1)^i}{n+k+i} \cdot \frac{(m+n+i)!}{i!(n-i)!(m+i)!}=0$

show that : $$\sum^{n}_{i=0} \frac{(-1)^i}{n+k+i} \cdot \frac{(m+n+i)!}{i!(n-i)!(m+i)!}=0$$ This problem is from : http://www.artofproblemsolving.com/Forum/viewtopic.php?p=239364&sid=2fbf367cb9fab8240df03e632a41085a#p239364 Evaluate a finite sum…
math110
  • 93,304
5
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2 answers

Evaluate $\sum_{n=2}^{\infty} \frac{1}{n\log_2^2{n}}$

I would like to evaluate $$\sum_{n=2}^{\infty} \frac{1}{n\log_{2}^{2}\left(n\right)}. $$ I know that $\int_2^{\infty} \frac{1}{n\log_2^2{n}} = \log{2}$ so my guess is that the answer is close to that. However numerically it seems to be very close to…
user115998
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