I was wondering how to generate the sum of infinite series for something simple like $$\sum_{x=1}^\infty 5^{-x}$$ I can't seem to figure out the logic as to why it comes out as 1/4 and I was unable to find a breakdown of the process that I would be capable of applying on the other simple similar problems.
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It's a geometric series. You can find formulas here : https://en.wikipedia.org/wiki/Geometric_series – mathreadler Feb 12 '17 at 19:29
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Hint:
$$S:=\frac15+\frac1{25}+\frac1{125}+\frac1{625}+\cdots$$ then
$$5S=1+\frac15+\frac1{25}+\frac1{125}+\cdots.$$
Make a second link with $S$ and conclude.
You can also work it out "experimentally", computing the successive terms by hand (this is not very difficult)
$$0.2\\+0.04=0.24\\+0.008=0.248\\+0.0016=0.2496\\+0.00032=0.24992\\+0.000064=0.249984\\+0.0000128=0.2499968\\\cdots$$
Then you can conjecture that the limit is $\dfrac14$.
To prove it more formally, compute the differences
$$\frac14-\frac15=\frac1{20},\\ \frac14-\frac15-\frac1{25}=\frac1{100},\\ \frac14-\frac15-\frac1{25}-\frac1{125}=\frac1{500},\\\cdots $$
You quickly observe the pattern, $\dfrac1{4\cdot5^n}$ and conclude that the partial sums are
$$\frac14-\frac1{4\cdot5^n}.$$
Then by induction
$$\left(\frac14-\frac1{4\cdot 5^n}\right)-\frac1{5^{n+1}}=\frac14-\frac1{4\cdot 5^{n+1}}.$$