All Questions
Tagged with summation algebra-precalculus
166
questions
135
votes
7
answers
107k
views
Values of $\sum_{n=0}^\infty x^n$ and $\sum_{n=0}^N x^n$
Why does the following hold:
\begin{equation*}
\displaystyle \sum\limits_{n=0}^{\infty} 0.7^n=\frac{1}{1-0.7} = 10/3\quad ?
\end{equation*}
Can we generalize the above to
$\displaystyle \sum_{n=...
68
votes
16
answers
54k
views
Proving $1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$ using induction
How can I prove that
$$1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$$
for all $n \in \mathbb{N}$? I am looking for a proof using mathematical induction.
Thanks
141
votes
36
answers
308k
views
Proof that $1+2+3+4+\cdots+n = \frac{n\times(n+1)}2$
Why is $1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2$ $\space$ ?
- 1,585
188
votes
28
answers
20k
views
Proving the identity $\sum_{k=1}^n {k^3} = \big(\sum_{k=1}^n k\big)^2$ without induction
I recently proved that
$$\sum_{k=1}^n k^3 = \left(\sum_{k=1}^n k \right)^2$$
using mathematical induction. I'm interested if there's an intuitive explanation, or even a combinatorial interpretation ...
- 5,877
18
votes
12
answers
17k
views
How to prove this binomial identity $\sum_{r=0}^n {r {n \choose r}} = n2^{n-1}$?
I am trying to prove this binomial identity $\displaystyle\sum_{r=0}^n {r {n \choose r}} = n2^{n-1}$ but am not able to think something except induction,which is of-course not necessary (I think) here,...
- 22.5k
18
votes
4
answers
13k
views
Formula for calculating $\sum_{n=0}^{m}nr^n$
I want to know the general formula for $\sum_{n=0}^{m}nr^n$ for some constant $r$ and how it is derived.
For example, when $r = 2$, the formula is given by:$$\sum_{n=0}^{m}n2^n = 2(m2^m - 2^m +1),$$...
- 2,475
24
votes
5
answers
10k
views
Algebraic Proof that $\sum\limits_{i=0}^n \binom{n}{i}=2^n$
I'm well aware of the combinatorial variant of the proof, i.e. noting that each formula is a different representation for the number of subsets of a set of $n$ elements. I'm curious if there's a ...
- 15.5k
35
votes
3
answers
4k
views
Trig sum: $\tan ^21^\circ+\tan ^22^\circ+\cdots+\tan^2 89^\circ = \text{?}$
As the title suggests, I'm trying to find the sum $$\tan^21^\circ+\tan^2 2^\circ+\cdots+\tan^2 89^\circ$$
I'm looking for a solution that doesn't involve complex numbers, or any other advanced branch ...
- 3,153
32
votes
5
answers
2k
views
How to prove $\sum\limits_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$?
Other than the general inductive method,how could we show that $$\sum_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$$
Apart from induction, I tried with Wolfram Alpha to check the validity, ...
- 22.5k
41
votes
5
answers
21k
views
How to find the sum of the series $1+\frac{1}{2}+ \frac{1}{3}+\frac{1}{4}+\dots+\frac{1}{n}$?
How to find the sum of the following series?
$$1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots + \frac{1}{n}$$
This is a harmonic progression. So, is the following formula correct?
$\frac{(number ~...
- 10k
4
votes
2
answers
4k
views
Proving the geometric sum formula by induction
$$\sum_{k=0}^nq^k = \frac{1-q^{n+1}}{1-q}$$
I want to prove this by induction. Here's what I have.
$$\frac{1-q^{n+1}}{1-q} + q^{n+1} = \frac{1-q^{n+1}+q^{n+1}(1-q)}{1-q}$$
I wanted to factor a $q^{...
- 493
116
votes
5
answers
126k
views
What is the term for a factorial type operation, but with summation instead of products?
(Pardon if this seems a bit beginner, this is my first post in math - trying to improve my knowledge while tackling Project Euler problems)
I'm aware of Sigma notation, but is there a function/name ...
- 1,389
10
votes
3
answers
354
views
Finding $\sum_{k=0}^{n-1}\frac{\alpha_k}{2-\alpha_k}$, where $\alpha_k$ are the $n$-th roots of unity
The question asks to compute:
$$\sum_{k=0}^{n-1}\dfrac{\alpha_k}{2-\alpha_k}$$
where $\alpha_0, \alpha_1, \ldots, \alpha_{n-1}$ are the $n$-th roots of unity.
I started off by simplifiyng and got ...
- 7,561
6
votes
7
answers
27k
views
Proof of the formula $1+x+x^2+x^3+ \cdots +x^n =\frac{x^{n+1}-1}{x-1}$ [duplicate]
Possible Duplicate:
Value of $\sum x^n$
Proof to the formula
$$1+x+x^2+x^3+\cdots+x^n = \frac{x^{n+1}-1}{x-1}.$$
- 81
31
votes
5
answers
13k
views
Is there any formula for the series $1 + \frac12 + \frac13 + \cdots + \frac 1 n = ?$
Is there any formula for this series?
$$1 + \frac12 + \frac13 + \cdots + \frac 1 n .$$
- 461