Linked Questions
86 questions linked to/from Values of $\sum_{n=0}^\infty x^n$ and $\sum_{n=0}^N x^n$
5
votes
7answers
2k 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
$$1+x+x^2+x^3+\cdots+x^n = \frac{x^{n+1}-1}{x-1}.$$
2
votes
5answers
316 views
Easy summation question: $S= 1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\frac{1}{16}\cdots$ [duplicate]
While during physics I encountered a sum I couldn't evaluate:
$$S= 1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\frac{1}{16}\cdots$$
Is there a particular formula for this sum and does it converges?
-4
votes
4answers
43k views
Proof of the power series 1 + $x^2$ + $x^3$ + $\ldots$ + $x^n$ = $\frac{1}{1-x}$ [duplicate]
Can anyone show me the proof of this equation:
$$
\lim_{n \to \infty} 1 + x + x^2 + x^3 + \ldots + x^n = \frac{1}{1-x},
$$
where $|x|<1$.
Edit: I have then additionally written $x$ on the left ...
2
votes
1answer
11k views
How to convert a series to an equation? [duplicate]
Possible Duplicate:
Value of $\sum\limits_n x^n$
I don't know the technical language for what I'm asking, so the title might be a little misleading, but hopefully I can convey my purpose to you ...
-1
votes
3answers
4k views
Why $ \sum_{k=0}^{\infty} q^k $ sum is $ \frac{1}{1-q}$ when $|q| < 1$ [duplicate]
Why is the infinite sum of $ \sum_{k=0}^{\infty} q^k = \frac{1}{1-q}$ when $|q| < 1$
I don't understand how the $\frac{1}{1-q}$ got calculated. I am not a math expert so I am looking for an easy ...
1
vote
4answers
470 views
Power summation of $n^3$ or higher [duplicate]
Possible Duplicate:
why is $\sum\limits_{k=1}^{n} k^m$ a polynomial with degree $m+1$ in $n$
If I want to find the formula for $$\sum_{k=1}^n k^2$$
I would do the next: $$(n+1)^2 = a(n+1)^3 + b(n+...
5
votes
5answers
443 views
Evaluating the sum of geometric series [duplicate]
Possible Duplicate:
Value of $\sum\limits_n x^n$
I'm trying to understand how to evaluate the following series:
$$
\sum_{n=0}^\infty {\frac{18}{3^n}}.
$$
I tried following this Wikipedia ...
0
votes
3answers
985 views
Prove that 1/2 + 1/4 + 1/8 … = 1 [duplicate]
I've often heard that instead of adding up to a little less than one, 1/2 + 1/4 + 1/8... = 1. Is there any way to prove this using equations without using Sigma, or is it just an accepted fact? I need ...
-1
votes
2answers
3k views
Simplifying the infinite series $\sum_{n = 1}^{\infty} \left(\frac{1}{2}\right)^{3n}$ [duplicate]
Does anyone know of a way to simplify
$$
\sum_{n = 1}^{\infty} \left(\frac{1}{2}\right)^{3n}
$$
to a number?
-4
votes
7answers
292 views
Find the sum of the geometric sequence [duplicate]
$$1+ \frac12 + \frac14+\dots+\frac1{2^n}$$
To find the sum of the equation you have to find $n$, the number of terms in the geometric sequence and I don't know how...
The answer in the text book is $...
0
votes
1answer
1k views
Conditions for convergence of a geometric series [duplicate]
This question concerns the infinite geometric series formula. It turns out there is a nice formula for the sum of an infinite geometric series.
Consider the infinite geometric series $1+r+r^2+r^3+\...
3
votes
2answers
156 views
$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots+\frac{1}{\infty}=?$ [duplicate]
One of my friends has given me the following problem:
$$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots+\frac{1}{\infty} = ?$$
He said that the answer is $1$. He has given his argument as that this is ...
-1
votes
4answers
147 views
How to calculate $k^0+k^1+k^2 + k^3+…+ k^{n-1}$ [duplicate]
How to simplify below expression or convert it to something simpler like $k^{n-1}$?
$$
k^0+k^1+k^2 + k^3+...+ k^{n-1}
$$
1
vote
3answers
135 views
Why is $\sum_{n=0}^{\infty }\left ( \frac{1}{2} \right )^{n}= 2$? [duplicate]
I'm sorry if this is duplicated, but I can not find any answer to it.
1
vote
4answers
131 views
What is the formula of: $a^{0} + a^{1} + a^{2} + … + a^{n-1} + a^{n}$? [duplicate]
What is the formula of:
$$a^{0} + a^{1} + a^{2} + ... + a^{n-1} + a^{n}$$
Any ideas?