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I'm hoping to get an expert opinion on the summation: $$\sum_{n=1}^{\infty} (-x)^n$$ I've attempted to find a closed-form of this expression as follows. $$\sum_{n=1}^{\infty} (-x)^n=S_n$$ $$S_n=-x+x^2+-x^3+x^4...$$ $$-xS_n=x^2+-x^3+x^4+-x^5...$$ $$S_n(1+x)=-x$$ $$S_n=\sum_{n=1}^{\infty} (-x)^n=\frac{-x}{x+1}$$ I'm curious whether there is a more rigorous way to derive this (involving more complex maths), or maybe to contradict my answer.

Another interesting thing is that the series $$\sum_{n=1}^{\infty} (x)^n=\frac{x}{1-x}$$ and when you plot both on a graph both they seem like rotations on one another ... but I don't know what do you guys think. Your input will be greatly appreciated.

Bernard
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    Your answer is correct. A more rigorous way would involve, probably, convergence radius (your solution is correct only for $;|x|<1$) . Perhaps you may want to read about Taylor and Maclaurin series. – DonAntonio Jul 26 '19 at 18:50
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    It's a geometric series. – Angina Seng Jul 26 '19 at 18:51
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    Clarifying DonAntonio's comment a bit, it's easier if your term $S_N$ is just a sum up to $N$ instead of $\infty$. Then using an algebraic rearrangement comparable to yours, you'd find that $S_N(1+x)=(-x)(1-(-x)^N)$. The $(1-(-x)^N)$ term only disappears under suitable conditions for $x$. – Jam Jul 26 '19 at 18:54
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    Also, using the variable $n$ in the term $S_n$ is slightly bad notation since $n$ was already used as an index in the sum so it can't be made into a variable. – Jam Jul 26 '19 at 18:56
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    There is an incredibly detailed similar answer for this question here: https://math.stackexchange.com/questions/29023/value-of-sum-limits-n-xn – wjmccann Jul 26 '19 at 18:56
  • $\sum_{n=1}^{\infty} x^n$ is equal to $\frac{x}{1-x}$, not $\frac{-1}{x-1}$. – TonyK Jul 26 '19 at 18:59
  • And then you just substitute $-x$ for $x$ in that formula to get $\sum_{n=1}^{\infty} (-x)^n=\frac{-x}{x+1}$. – TonyK Jul 26 '19 at 19:00
  • and $\dfrac{-x}{1-(-x)}=\dfrac{-x}{x+1}$ explains symmetry on the graph – J. W. Tanner Jul 26 '19 at 19:34

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