Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [geometric-series]

For questions about or involving geometric series, a series where successive terms have a common ratio.

A geometric series is of the form

$$\sum_{n=0}^{\infty}ar^n.$$

If $|r| < 1$, then the series converges to $\frac{a}{1-r}$. If $|r| \geq 1$, the series diverges.

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How can I evaluate $\sum_{n=0}^\infty(n+1)x^n$?

How can I evaluate $$\sum_{n=1}^\infty\frac{2n}{3^{n+1}}$$? I know the answer thanks to Wolfram Alpha, but I'm more concerned with how I can derive that answer. It cites tests to prove that it is convergent, but my class has never learned these…
backus
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Evaluate $\lim\limits_{n\rightarrow \infty}\frac{n+n^2+n^3+\cdots +n^n}{1^n+2^n+3^n+\cdots +n^n}.$

Problem Evaluate $$\lim\limits_{n\rightarrow \infty}\frac{n+n^2+n^3+\cdots +n^n}{1^n+2^n+3^n+\cdots +n^n}.$$ My solution Notice that $$\lim_{n \to \infty}\frac{n+n^2+n^3+\cdots +n^n}{n^n}=\lim_{n \to \infty}\frac{n(n^n-1)}{(n-1)n^n}=\lim_{n \to…
WuKong
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What is wrong with the sum of these two series?

Could anyone help me to find the mistake in the following problem? Based on the formula of the sum of a geometric series: \begin{equation} 1 + x + x^{2} + \cdots + x^{n} + \cdots = \frac{1}{1 - x} \end{equation} \begin{equation} 1 + \frac{1}{x} +…
Kikolo
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Geometric series with arbitrary signs

Suppose $S_n = \sum_{i=0}^n c_i \alpha^i$, where $c_n \in \{ 1,-1\} $ for all $n \geq 0$, and $\alpha > 1$. I want to show that $|S_n| \to \infty$. For $\alpha > 2$, it easily follows from the triangle inequality, $$|\sum_{i=0}^n c_i \alpha^i| \geq…
student007
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Crazy Iterated Square Roots

I was messing around with infinite square root nesting problems like $$w_1=\sqrt{1+\sqrt{1+\sqrt{...}}}$$ which is an easy example. I decided to try one where the terms inside of the square roots form a geometric…
Franklin Pezzuti Dyer
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Is there a relationship between $\sum _{n=1}^{\infty }\left({\frac {1}{2}}\right)^{n} = 1$ and $\int_{1}^{\infty} \frac{1}{x^2} \,dx = 1$?

A classic example of an infinite series that converges is: $${\displaystyle {\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+\cdots =\sum _{n=1}^{\infty }\left({\frac {1}{2}}\right)^{n}=1}$$ A classic example of an infinite integral that…
Toph
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A tough integral $\int_0^{\infty}\frac{\operatorname{sech}(\pi x)}{1+4x^2}\, \mathrm dx $

I was working on an integral which I found on Quora. I simplified it a lot and ended up with this intgeral $$\int_0^{\infty}\dfrac{\operatorname{sech}(\pi x)}{1+4x^2}\, \mathrm dx $$ I tried converting this into exponential form and using geometric…
Laxmi Narayan Bhandari
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Is $1111111111111111111111111111111111111111111111111111111$ ($55$ $1$'s) a composite number?

This is an exercise from a sequence and series book that I am solving. I tried manipulating the number to make it easier to work with: $$111...1 = \frac{1}9(999...) = \frac{1}9(10^{55} - 1)$$ as the number of $1$'s is $55$. The exercise was under…
user69284
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Is there a non-trivial arithmetic progression of positive integers such that every number contains the digit $2?$

Let $X:=\{$ positive integers that contain the digit $2\}$ For fixed $m,n\in\mathbb{N},$ define the A.P. $S_{m,n:=}\ \{m,\ m+n,\ m+2n, \ldots\}\ .$ I am interested in $S_{m,n}\cap X,$ and $S_{m,n}\cap (\mathbb{N}\setminus X).$ It is clear that $…
Adam Rubinson
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How to find the sum of $1+(1+r)s+(1+r+r^2)s^2+\dots$?

I was asked to find the geometric sum of the following: $$1+(1+r)s+(1+r+r^2)s^2+\dots$$ My first way to solve the problem is to expand the brackets, and sort them out into two different geometric series, and evaluate the separated series…
Loo Soo Yong
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Formula for finite power series

Are there any formula for result of following power series? $$0\leq q\leq 1$$ $$ \sum_{n=a}^b q^n $$
Muaa2404
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Intuition behind the finite geometric series formula?

Can anyone give some intuition or insight on why $S_n = a(\frac{1-r^n}{1-r})$ works? (I've seen the proof but I like being able to visualize to think about formulas in different ways.)
Jack Pan
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Relationship between taylor series and geometric series

To find the taylor series of a function you would usually use the formula $\sum_{n=0}^{\infty}\frac{f^{n}(c)}{n!}(z-c)^n$. However when computing the taylor series for $f(z)=\frac{1}{z+3}$ about $z=1$, I discovered that not only can you compute it…
user342661
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When is a Power Series a Geometric Series?

A geometric series is one having a common ratio, right? Something like: $$\sum_{n=0}^{\infty} ar^n$$ And a power series is one with the form: $$\sum_{n=0}^{\infty} c_n(x-a)^n$$ Initially, I thought a geometric power series was one which had a…
tommytwoeyes
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Longest geometric progression of primes

There are arbitrarily long arithmetic progressions of primes e.g. $5, 11, 17, 23, 29$ for a $5$-length progression, but no (infinite) arithmetic sequence of primes with common difference $d\neq 0$, as $d\in\mathbb{Z}$ is an obvious constraint and…
A.M.
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