Is there a reason why a series of the form $1+2+4+8+16+...$ called a Geometric series and another of the form $1+3+5+7+9+...$ called an arithmetic series?
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Where does the name of these two famous types of sequences come from?  The article  of Wikipedia says that the geometric sequence is called like this because every term is the geometric mean of its two adjacent terms. Though it is true, it only reduces the question to: why is the geometric mean geometric (in opposition to the arithmetic mean).   Continuing my investigation on , I was told that a square with the same area than a rectangle with sides $a$ and $b$ has their geometric mean $\sqrt{ab}$ for side. That's again totally true, but a square with same perimeter than this rectangle of sides $a$ and $b$ has their arithmetic mean $\frac{a+b}{2}$ for side!  Thus, my question: Who coined these names? And why? Why is the geometric mean more geometric than the arithmetic mean?
113346
I know the series, $4-{4\over3}+{4\over5}-{4\over7}...$ converges to $\pi$ but I have heard many people say that while this is a classic example, there are series that converge much faster.  Does anyone know of any?
46970
I puzzled two high school Pre-calc math teachers today with a little proof (maybe not) I found a couple years ago that infinity is equal to -1:   Let x equal the geometric series: $1 + 2 + 4 + 8 + 16 \ldots$  $x = 1 + 2 + 4 + 8 + 16 \ldots$ Multiply each side by 2:  $2x = 2 + 4 + 8 + 16 + 32 \ldots$ Again from the equation in step 1, move the $1$ term to the left hand of the equation:  $x - 1 = 2 + 4 + 8 + 16 + 32 \ldots$ So the following appears to be true:  $2x = x - 1 \implies x = -1$   This is obviously illogical. The teachers told me the problem has to do with adding the two infinite geometric series, but they weren't positive. I'm currently in Pre-calc, so I have extremely little knowledge on calculus, but a little help with this paradox would be appreciated.
105777
I need to find the sum of this series: $1+\frac 12 + \frac 13 + \cdots + \frac 1{50}$   Please help me find the sum of this series.
80866
$\text{What are some interesting cases of $\pi$ appearing in situations that do not seem geometric?}$  Ever since I saw the identity $$\displaystyle \sum_{n = 1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$$  and the generalization of $\zeta (2k)$, my perception of $\pi$ has changed. I used to think of it as rather obscure and purely geometric (applying to circles and such), but it seems that is not the case since it pops up in things like this which have no known geometric connection as far as I know. What are some other cases of $\pi$ popping up in unexpected places, and is there an underlying geometric explanation for its appearance?  In other words, what are some examples of $\pi$ popping up in places we wouldn't expect?
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I have a clueless friend who believes that   $$ \sum_{n=1}^\infty \frac{1}{2^n} $$  doesn't equal $1$ in the 'normal arithmetical sense'.   He doesn't believe that this series flat out equals $$1$$ Is he correct?
30562
Show the following series converges     $$ \sum_{j=1}^{\infty} \frac{(2^j)+ j}{(3^j) - j} . $$   I tried to use the comparison test and tried to compare it with the series of $\dfrac{2^j}{3^j}$ because this is the geometric series. However, $\dfrac {2^j}{3^j}$ is smaller than $\dfrac{2^j+j}{3^j-j}$, so by comparison test, the series $\sum_{j=1}^{\infty}\dfrac{2^j}{3^j}$ converges does NOT indicate that the series $\sum_{j=1}^{\infty}\dfrac{2^j+j}{3^j-j}$ converges... Thus I'm confused. Thanks!  Thanks!
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I want to find a formula for the sum of this series using its general term. How to do it?  Series $$ S_n = \underbrace{1/3 + 2/21 + 3/91 + 4/273 + \cdots}_{n \text{ terms}} $$  General Term  $$ S_n = \sum_{i=1}^{n} \frac{i}{i^4+i^2+1} $$
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I'm a bit confused as to this problem:  Consider the infinite series:  $$\left(1 - \frac 12\right) + \left(\frac 12 - \frac 13\right) + \left(\frac 13 - \frac 14\right) \cdots$$  a) Find the sum $S_n$ of the first $n$ terms.  b) Find the sum of this infinite series.  I can't get past part a) - or rather I should say I'm not sure how to do it anymore.  Because the problem asks for a sum of the infinite series, I'm assuming the series must be Geometric, so I tried to find the common ratio based on the formula  $$r = S_n / S_{n-1} = \frac{\frac 12 - \frac 13}{1 - \frac 12} = \frac 13$$  Which is fine, but when I check the ratio of the 3rd and the 2nd terms:  $$r = \frac{\frac 13 - \frac 14}{\frac 12 - \frac 13} =\frac 12$$  So the ratio isn't constant... I tried finding a common difference instead, but the difference between 2 consecutive terms wasn't constant either.  I feel like I must be doing something wrong or otherwise missing something, because looking at the problem, I notice that the terms given have a pattern:  $$\left(1 - \frac 12\right) + \left(\frac 12 - \frac 13\right) + \left(\frac 13 - \frac 14\right) + \cdots + \left(\frac 1n - \frac{1}{n+1}\right)$$  Which is reminiscent of the textbook's proof of the equation that yields the sum of the first $n$ terms of a geometric sequence, where every term besides $a_1$ and $a_n$ cancel out and yield  $$a_1 \times \frac{1 - r^n}{1 - r}.$$  But I don't know really know how to proceed at this point, since I can't find a common ratio or difference.
97514
I am trying to sum the following series, given n is a positive integer, $$ n+n(n-1)+n(n-1)(n-2)+...+n! $$ I think there is a solution, but if not, an approximate result will also be appreciated.   Regards.
82707
Consider the following function:  $$S(n)=1+ c + c^2 + ··· + c^n,$$  where c is a positive real number.  (A) This function is the sum of a geometric series. Give a precise closed-form formula for $S(n)$, interms of $c$ and $n$, in the case where $c \neq 1$.  (B) Show that $S(n)$  is:  $\theta(1)$ if $c&lt;1$  $ \theta(n)$ if $c=1$  $\theta(c^n)$ if $c&gt;1$    can someone please explain how i can go about finding the solutions for these problems, i have never encountered these and am stuck, i know that big theta means that the function is bond on both sides but how do i got about finding the answer?  (edit) im still having trouble understanding part B. can someone please explain? (can someone walk me through the second one) so far i have $n \leq c^{n+1} \leq ???$  (edit #2) i tried doing the first one can someone verify if i did it right  $S(n)$ is $\Theta(1)$ if c&lt;1  (1)$S(n)$ is $O(1)$ if c&lt;1  (2)$S(n)$ is $\Omega(1)$ if c&lt;1  (1) $$\frac{C^{n+1}-1}{C-1} \lt (C_1)(1)$$ $$\frac{C^{n+1}-1}{C-1} \lt (C_1)$$ $$C^{n+1} \lt C_1(C-1)$$ $$C^nC-1 \lt C_2C-1$$ $$C^n \lt C_2$$ $$C^n \lt 1$$ since c&lt;1 this means that any number that will be $C^n$ will be less then 1 for all positive n  i did the omega part pretty much the same way and got $$C^n \gt C_2$$ $$C^n \gt 0$$ since c&lt;1, $C^n$ can potentially become very very small so the only thing bounding it is 0 for all positive n
44381
My problem is to find the sum of the series $$ S = 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{6}+\frac{1}{8}+\frac{1}{9}+\frac{1}{12}+\cdots $$ where the terms are the reciprocals of the positive integers whose only prime factors are $2$s and $3$s.   I can see that every term in $S$ is of the form $\dfrac{1}{2^m3^n}$ where $m,n\geq 0$. I can also see that each term only occurs once, but I'm not really sure how to use this information effectively. Any ideas (apparently geometric series was a hint)?
68121
Does the following series converge or diverge?  $\sum_{n=1}^{\infty} a_n$ where  $a_n = \frac 1 b_n$, and $(b_n)_n$ is the subsequence of $(n)_n$ whose terms do not have a $9$ in their decimal notation.  I have no idea how the sequence $b_n$ even looks like. If I have an idea of how to prove that a series converges or not with criteria but I don't if I don't know what $b_n$ is.
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I'm not sure if I should call this "the sum of a sum" but I can't think of another phrase for it.   Anyhow, I'm confused about excercises such as this one  $$\sum_{j=2}^{\infty} \left( \sum_{k=2}^{\infty} k^{-j} \right)$$  where I want to compute the sum. I'm thinking maybe I could rewrite it as  $$\sum_{k=2}^{\infty} \frac{1}{k^{2}} + \sum_{k=2}^{\infty} \frac{1}{k^{3}}+... $$  but I don't know where to go from there or if that is even correct thinking. (I just began my first course in linear analysis.)
47326
How do you prove $$\sum_{n=1}^{\infty} \frac {n}{2^n} = 2\ ?$$  My attempt: I have been trying to find geometric series that converge to 2 which can bind the given series on either side. But I am unable to find these. Is there a general technique to find the sum? This is a high school interview question and must be easy enough to solve in a few minutes.  Please give any hints for the first step towards a solution.
41670
I'm working on my physics master course homework and I'm given the following equation out of nowhere:  $\displaystyle{ 1 + \sum_{n\ =\ 1}^{\infty}{z^n\left(\, 2n - 1\,\right)!! \over 2n!!} ={1 \over \,\sqrt{\,\vphantom{\large A}1 - z\,}\,} }$  Now I don't need to prove it for my Homework, but still im wondering, how one would calculate this series. Of course it is not mentioned in the Homework, that this series doesn't converge for all $z$ ( sloppy physicsy style, I know : ) ). Wolfram Alpha said, It only converges for  $\,{\rm abs}\left(\, z\,\right) &lt; 1$, which is a hint to the geometric series, but I have no idea how to account for the double factiorals.  Any hints?
68700
I came across this when learning about sequences and series. No one proved it to me tho! Is there a link anywhere? Or would it be beyond what I know? (First term of analysis)