New formula about Geometric series with altering signs

I don't know maybe someone worked with such geometric series and created formulas but although I researched so much on the web I haven't seen anything.If anyone interested in I can send algebraic proofs and we can work together.Since I am not a math expert I cannot proof further such as specific number set but I think the ''X'' can be in real numbers set I tried a lot of numbers also I used calculators It is working correctly

Note by Hakan Eskici
1 year, 9 months ago

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Proof: S=1+x+x2+...+xnS = 1+x+x^2+...+x^n S×x=x+x2+...+xn+1S \times x = x+x^2+...+x^{n+1} SS×x=S(1x)=1xn+1S - S \times x = S(1-x) = 1 - x^{n+1} S=1xn+11xS = \frac{1 - x^{n+1}}{1-x} You can choose x < 0; also generally (1)k×ak=(a)k(-1)^k \times a^k = (-a)^k Because I can see negative signs in your solution I assume that you want to choose x >= 0. S=1xn+11xS=\frac{1 - x^{n+1}}{1-x} S=1(x)n+11(x)S'=\frac{1 - (-x)^{n+1}}{1-(-x)} S=1(x)n+11+xS'=\frac{1 - (-x)^{n+1}}{1+x} Now we multiply the denominator and the enumerator by x and get: S=x+(x)n+2x+x2S'=\frac{x + (-x)^{n+2}}{x+x^2} Q.E.D. (Note that the - changes to a + because the exponent of the -1 is raised by 1)

Poca Poca - 1 year, 9 months ago

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Note that for x=0; x=+/-1 the series is not a geometric series.

Poca Poca - 1 year, 9 months ago

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Great work!

Pi Han Goh - 1 year, 9 months ago

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Thank you so much.You approached with more elegant way.

Hakan Eskici - 1 year, 9 months ago

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But does it work for x=0x = 0 or for x=1x=-1?

Pi Han Goh - 1 year, 9 months ago

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No, I don't think so until now only they are the exceptions.

Hakan Eskici - 1 year, 9 months ago

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In general, can you find a simple expression of a+ar+ar2++arn1a + ar + ar^2 + \cdots + ar^{n-1} ?

Pi Han Goh - 1 year, 9 months ago

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@Pi Han Goh no, you can't signs must be altering like in the picture

Hakan Eskici - 1 year, 9 months ago

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@Hakan Eskici Are you saying that if r<0r < 0 , then we can't find a simple expression of a+ar++arn1a + ar + \cdots + ar^{n-1} ?

Pi Han Goh - 1 year, 9 months ago

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@Pi Han Goh oh sorry, you see why I am asking for help :D,it works with negative numbers even with irrational or rational ones.Why you aren't testing ?

Hakan Eskici - 1 year, 9 months ago

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@Hakan Eskici I'm trying to tell you that all your equations came from the same formula.

Pi Han Goh - 1 year, 9 months ago

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@Pi Han Goh hmm, what about if r>0r > 0 normal geometric series formula cannot provide it

Hakan Eskici - 1 year, 9 months ago

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@Hakan Eskici It works too.

Pi Han Goh - 1 year, 9 months ago

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@Pi Han Goh What should I do now can you give advice?

Hakan Eskici - 1 year, 9 months ago

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@Hakan Eskici In general, can you find a simple expression of a+ar+ar2++arn1a + ar + ar^2 + \cdots + ar^{n-1} ?

Hint: Let SS denote the value of this expression. What is SSrS - Sr ?

Pi Han Goh - 1 year, 9 months ago

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@Pi Han Goh yes we can with this formula [a(1 - rn )/ 1 - r].Have you been asking me this for the whole time? I thought we were talking about my formula.Okay, I am listening.

Hakan Eskici - 1 year, 9 months ago

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