Geometric sequences; partial sums with sum notation going from n=1 to infinity.

Question

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Value of $\sum\limits_n x^n$

I have been working on this problem for awhile and have made efforts to try to resolve it. The question is: $$\sum_{n=1}^\infty 7(0.1)^{n−1}$$

Answer 1

For the "partial sum", I assume you meant $\sum_{n=1}^k 7(0.1)^{n−1}$ for any positive integer $k$.

Taking the $7$ out of the sum, and shifting the exponent, we get $7\sum_{n=0}^{k-1} 0.1^n$.

All this needs now is the standard formula $\sum_{n=0}^{k-1} x^n = (x^k-1)/(x-1) = (1-x^k)/(1-x)$. Putting $x = 0.1$, $\sum_{n=0}^{k-1} 0.1^n = (1-0.1^k)/0.9$.