# Questions tagged [summation]

11,405 questions
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### amortized analysis

a) define f(k) as the largest power of 2 that divides k. For example, f(25) = 1, f(42) = 2, f(144) = 16. What is ${1 \over k}\sum_1^k f(k)$? b) define f(k) as the square of largest power of 2 that ...
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### Binomial series with two binomial coefficents

My question reads: Does this formula has mathematical meaning at first place? Is it summable? $$\sum^{\infty}_{k=0}{n\choose k}{m\choose k} x^k$$
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### Sum of the Reciprocal of the Difference of Two Squares

Is there a closed-form expression for the sum: $$\sum \limits_{i=2}^{n} \frac{1}{i^2-1}$$
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### Working on this summation proof

$$\sum_{i=1}^n t^{i-1}$$ I am stuck with the proof of this equality.
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### Prove by induction $\sum_{i=1}^ni^3=\frac{n^2(n+1)^2}{4}$ for $n\ge1$ [duplicate]

Prove the following statement $S(n)$ for $n\ge1$: $$\sum_{i=1}^ni^3=\frac{n^2(n+1)^2}{4}$$ To prove the basis, I substitute $1$ for $n$ in $S(n)$: $$\sum_{i=1}^11^3=1=\frac{1^2(2)^2}{4}$$ Great. ...
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### Proving $\sum\limits_{i=1}^{n}{\frac{i}{2^i}}=2-\frac{n+2}{2^n}$ by induction.

Theorem (Principle of mathematical induction): Let $G\subseteq \mathbb{N}$, suppose that a. $1\in G$ b. if $n\in \mathbb{N}$ and $\{1,...,n\}\subseteq G$, then $n+1\in G$ ...
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### Smallest eigenvalues of Sum of Two Positive Matrices

Let $C = A + B$, where $A$, $B$, and $C$ are positive definite matrices. In addition, $C$ is fixed. Let $\lambda (A)$, $\lambda (B)$, and $\lambda (C)$ be smallest eigenvalues of $A$, $B$, and $C$, ...
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### How to prove by induction that $\sum^n_{i=1}2^{i-1}=2^n-1$? [duplicate]

Possible Duplicate: How do I prove this by induction? (sum of powers of 2) Summation equation for $2^{x-1}$ How can I prove the following by induction? $$\sum^n_{i=1}2^{i-1}=2^n-1$$ I get to ...
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### Complex analysis equality in a limited sum

Let $z=e^{i\theta}$ with $\theta \in [0,2\pi[$ . Consider the sum $$\sum_{n=1}^{N} (e^{i\theta})^n.$$ How could this be equal to $$\frac{1-e^{iN+T\theta}}{1-e^{i\theta}} \quad ?$$ I tried to ...
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### Is it possible to make $\sum_{k=0}^{n-i}10^k \binom{n-i}{k}$ shorter?

I've got some expression $$\sum_{i=1}^n\left( 2^{i-1} a_i \sum_{k=0}^{n-i}10^k \binom{n-i}{k} \right)$$ $a_i$ stands for i-th digits in number. I came up with this formula from task in which we ...
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### Prove the following identity without using induction: $\sum\limits_{i=0}^n a^i = \frac{1-a^{n+1}}{1-a}$ for $a\neq1$

I'm struggling with proving $$\sum^n_{i=0}a^i = \frac{1-a^{n+1}}{1-a}$$ for $a\neq1$, not sure where to start.
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### How can the Möbius function be applied to a series?

Given a series $p_n(s)=\sum_{k=1}^n a_k$. I'd like to get $\hat{p}_n(s)=\sum_{k=1}^n \mu(k)a_k$. Think of $a_k=k^{-s}$ for example. If you let $n$ go to $\infty$, you'll see the well-known relation ...
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### Simplify $\sum_{i=0}^{n-1} { {2n}\choose{i}}\cdot x^i$

I am trying to simplify an expression involving summation as follows: $$\sum_{i=0}^{n-1} { {2n}\choose{i}}\cdot x^i$$ where $n$ is an integer, and $x$ is a positive real number. At a first glance,...
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### Evaluate $\sum_{n=1}^{1024}\left \lfloor \log_2n \right \rfloor$.

Evaluate $\sum_{n=1}^{1024}\left \lfloor \log_2n \right \rfloor$. I thought the answer is $1+1*2+2*2^2+3*2^3+4*2^4+5*2^5+6*2^6+7*2^7+8*2^8+9*2^9+2^{10}=9219$, but the answer should be 8204. What ...
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### Finding $\frac{3}{1!+2!+3!}+\frac{4}{2!+3!+4!} +\frac{5}{3!+4!+5!}+\cdots+\frac{2008}{2006!+2007!+2008!}$ [duplicate]

How to find : $$\frac{3}{1!+2!+3!}+\frac{4}{2!+3!+4!} +\frac{5}{3!+4!+5!}+\cdots+\frac{2008}{2006!+2007!+2008!}$$
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### Combinatorially prove that $\sum_{i=0}^n {n \choose i} 2^i = 3^n$

So i'm not sure at all how to prove things using a combinatorial proof. Where to do i start? What do i need to think about etc. For example how would i prove $$\sum_{i=0}^n {n \choose i} 2^i = 3^n$$...
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### Finding $\sum\limits_{n=1}^{9999} \frac{1}{(\sqrt{n+1}+\sqrt{n})(\sqrt[4]{n}+\sqrt[4]{n+1})}$

How we can find $$\sum_{n=1}^{9999} \frac{1}{(\sqrt{n+1}+\sqrt{n})(\sqrt[4]{n}+\sqrt[4]{n+1})}$$
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### Is there a direct, elementary proof of $n = \sum_{k|n} \phi(k)$?

If $k$ is a positive natural number then $\phi(k)$ denotes the number of natural numbers less than $k$ which are prime to $k$. I have seen proofs that $n = \sum_{k|n} \phi(k)$ which basically ...
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### Evaluate the following limit: $\lim_{n\to\infty}\left(\sum_{r=1}^{n}{\frac{r}{n^{2}+n+r}}\right)$

Evaluate the following limit: $$\lim_{n\to\infty}\left(\sum_{r=1}^{n}{\frac{r}{n^{2}+n+r}}\right)$$ The answer given is $\frac{1}{2}$.
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### Evaluation of $\sum_{k=0}^n \cos k\theta$

I just wanted to evaluate $$\sum_{k=0}^n \cos k\theta$$ and I know that it should give $$\cos\left(\frac{n\theta}{2}\right)\frac{\sin\left(\frac{(n+1)\theta}{2}\right)}{\sin(\theta / 2)}$$ ...
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### Help evaluating a limit

I have the following limit: $$\lim_{n\rightarrow\infty}e^{-\alpha\sqrt{n}}\sum_{k=0}^{n-1}2^{-n-k} {{n-1+k}\choose k}\sum_{m=0}^{n-1-k}\frac{(\alpha\sqrt{n})^m}{m!}$$ where $\alpha>0$. ...
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### Summation of series of product of Fibonacci numbers

What is the sum of following product of Fibonacci numbers $$\sum_{k=1}^{n-1} Fib(k)*Fib(n+3-k)$$ can anyone suggest only approach to find general term?
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### Sigma notation only for odd iterations

$\sum_{i=0}^{5}{i^2} = 0^2+1^2+2^2+3^2+4^2+5^2 = 55$ How to write this Sigma notation only for odd numbers: $1^2+3^2+5^2 = 35$ ?
### The number of ways to represent $(n^2+n)/4$ as a sum of $n/2$ distinct integers in $1,\dots,n$
For any positive integer $n$ (using integer division only), let $P(n)$ denote the number of ways in which the integer $(n^2+n)/4$ can be expressed as a sum of exactly $n /2$ distinct elements of the ...