# Questions tagged [geometric-series]

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

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### How to calculate the annualized geometric return from monthly data point and also have empty months

I have following series of monthly returns with empty months (no investment made). JanX1 --> 0.0125% FebX1 --> 0.009% MarX1 --> 0.024% AprX1 --> 0.047% MayX1 --> 0....
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### Forms of Geometric Series

I came across three forms of geometric series and all make sense with the exception of one: $\sum_{n=1}^{\infty}ar^{n+1}$ If the first term of the geometric series has to be a, how does n start at 1 ...
1 vote
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### Analytic continuation of a lambert series $f(z)=\sum_{n=1}^\infty \frac{2^n z^n}{1-(z/2)^n}$?

Let's define a function for complex $z$ : $$f(z)=\sum_{n=1}^\infty \frac{2^n z^n}{1-(z/2)^n}$$ Lambert series often have an analytic continuation and can even be entire functions so the poles can be ...
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1 vote
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### geometric series with negative exponent

I wonder if it is possible to convert the infinite geometric series with negative exponent to a positive one? Is the calculation is correct ? ∑_(n=1)^∞▒x^(-n) =∑_(n=1)^∞▒〖(〖1/x)〗^n 〗=1/(1-1/x)=x/(x-1)...
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### Is this possible to prove statement for any square matrix

I have a statement: $det(I + A + A^2 + ... + A^{2022}) \ge 0$, where A is a square matrix. The task is to prove it and say, is it possible for it to be equal to $0$? I'm preparing for the math contest ...
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### Series approximation to $\frac{1}{1-(1+x)^{-n}}$

I'm attempting to find a series approximation to: $$f(x)=\frac{1}{1-(1+x)^{-n}}$$ where $n\in\mathbb{N^+}$ and is a constant, and $x\in\mathbb{R}$ and $0<x<1$. Using Wolfram Alpha, I noted the ...
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### Closed form of a geometric series without some term

I have to study find the closing form of the following series: \begin{equation} f(1) + f(2) + f(3) + f(5) + f(6) + f(7) + \dots \end{equation} So basically the sum of all terms without the multiple of ...
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### Any idea on how WolframAlpha did this sum?

So I've been trying to work out this sum for quite a while: WolframAlpha unfortunately won't supply step by step proofs for this for some reason... As for how i tried to prove this: I looked at sin((...
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### How do you compute the result of an equation with discrete values

A bit of background - I had a phonecall appointment with the drs and it wasn't until a few hours after the appointment time that I received the call. I decided to try and model how much extra time I ...
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1 vote
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### Probability of printing without a skip.

The old printer in the computer room skips, on average, 1 character in 25 (i.e., the chance of skipping any particular character is 1/25). First part : Determine the probability that a line of 45 ...
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### Finding $S_n$ in terms of n for the sequence (6 + 66 + 666 + 6666 ...)

Sequence given : 6, 66, 666, 6666. Find $S_n$ in terms of n The common ratio of a geometric progression can be solved is $\frac{T_n}{T_{n-1}} = r$, where r is the common ratio and n is the When ...
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