I am trying to understand power series, could someone review this exercise? So basically I need to find the sum of the following power series$$\sum_{n=0}^{\infty}(-1)^n(4x)^n$$ So basically I need to use this formula and do some substitutions$$\frac{1}{1-x} = \sum_{n=0}^{\infty}x^n$$ So if we substitute $x = -4t$ we get $$\frac{1}{1+4t}=\sum_{n=0}^{\infty}(-4t)^n$$ which is kinda what I want right? So, can I conclude that $$\frac{1}{1+4x}=\sum_{n=0}^{\infty}(-1)^n(4x)^n$$? Now, supposing that this is the correct result(which probably isn’t) how can I find the sum of this series. Could someone show me a general approach for finding the sum of a series?. Thank you in advance.
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You did find the correct expression. Just substitute the value of x to whatever you want, now. – Ritam_Dasgupta Sep 19 '21 at 08:37
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@Ritam_Dasgupta thank you! – Alex D'ago Sep 19 '21 at 08:43
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Your answer is correct. This is a geometric series with common ratio $r=-4$ and coefficient of each term $a=1$. You can find some derivations here. You may also find this useful.
Note that we require that $|x|<1$ for the series $\sum_{n=0}^{\infty}x^{n}=\frac{1}{1-x}$ to converge. Thus we have
$$\sum_{n=0}^{\infty}(-1)^{n}(4x)^n=\frac{1}{1+4x}$$
which converges when $|x|<\frac{1}{4}$.
Alessio K
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