\(A=\left(-\dfrac{1}{7}\right)^0+\left(\dfrac{-1}{7}\right)^1+...+\left(-\dfrac{1}{7}\right)^{2007}\)

\(\Leftrightarrow\left(-\dfrac{1}{7}\right)A=\left(-\dfrac{1}{7}\right)^1+\left(-\dfrac{1}{7}\right)^2+...+\left(-\dfrac{1}{7}\right)^{2008}\)

\(\Leftrightarrow-\dfrac{8}{7}A=\left(-\dfrac{1}{7}\right)^{2008}-1\)

\(\Leftrightarrow A=\left(\dfrac{1}{7^{2008}}-\dfrac{7^{2008}}{7^{2008}}\right):\dfrac{-8}{7}=\dfrac{1-7^{2008}}{7^{2008}}\cdot\dfrac{-7}{8}=\dfrac{7^{2008}-1}{8\cdot7^{2007}}\)

Đặt \(A=\left(\frac{-1}{7}\right)^0+\left(\frac{-1}{7}\right)^1+\left(\frac{-1}{7}\right)^2+...+\left(\frac{-1}{7}\right)^{2007}\)

\(\frac{-1}{7}.A=\left(\frac{-1}{7}\right)^1+\left(\frac{-1}{7}\right)^2+\left(\frac{-1}{7}\right)^3+...+\left(\frac{-1}{7}\right)^{2008}\)

\(A-\frac{-1}{7}.A=\left[\left(\frac{-1}{7}\right)^0+\left(\frac{-1}{7}\right)^1+\left(\frac{-1}{7}\right)^2+...+\left(\frac{-1}{7}\right)^{2007}\right]-\left[\left(\frac{-1}{7}\right)^1+\left(\frac{-1}{7}\right)^2+\left(\frac{-1}{7}\right)^3+...+\left(\frac{-1}{7}\right)^{2008}\right]\)

\(A+\frac{1}{7}.A=\left(\frac{-1}{7}\right)^0-\left(\frac{-1}{7}\right)^{2008}\)

\(\frac{8}{7}.A=1-\left(\frac{1}{7}\right)^{2008}\)

\(\frac{8}{7}.A=1-\frac{1}{7^{2008}}\)

\(A=\left(1-\frac{1}{7^{2008}}\right):\frac{8}{7}=\frac{\left(1-\frac{1}{7^{2008}}\right).7}{8}\)

:) 

S=(-1/7)⁰+(-1/7)¹+...+(-1/7)²⁰⁰⁷

-1/7.S=(-1/7)¹+(-1/7)²+...+(-1/7)²⁰⁰⁸

-1/7.S-S=[(-1/7)¹+(-1/7)²+...+(-1/7)²⁰⁰⁸]-[(-1/7)⁰+(-1/7)¹+...+(-1/7)²⁰⁰⁷]

-8/7.S=(-1/7)²⁰⁰⁸-(-1/7)⁰

-8/7.S=(1/7)²⁰⁰⁸-1

.........................

\(S=\left(-\dfrac{1}{7}\right)^0+\left(-\dfrac{1}{7}\right)^1+\left(-\dfrac{1}{7}\right)^2+...+\left(-\dfrac{1}{7}\right)^{2017}\\ S=\dfrac{\left(-1\right)^0}{7^0}+\dfrac{\left(-1\right)^1}{7^1}+\dfrac{\left(-1\right)^2}{7^2}+...+\dfrac{\left(-1\right)^{2017}}{7^{2017}}\\ S=\dfrac{1}{7^0}+\dfrac{-1}{7^1}+\dfrac{1}{7^2}+...+\dfrac{-1}{7^{2017}}\\ -7S=\dfrac{-7}{7^0}+\dfrac{7}{7^1}+\dfrac{-7}{7^2}+...+\dfrac{7}{7^{2017}}\\ -7S=\left(-7\right)+\dfrac{1}{7^0}+\dfrac{-1}{7^1}+...+\dfrac{1}{7^{2016}}\\ -7S-S=\left[\left(-7\right)+\dfrac{1}{7^0}+\dfrac{-1}{7^1}+...+\dfrac{1}{7^{2016}}\right]+\left(\dfrac{1}{7^0}+\dfrac{-1}{7^1}+\dfrac{1}{7^2}+...+\dfrac{-1}{7^{2017}}\right)\\ -8S=\left(-7\right)+\dfrac{-1}{2017}\\ -8S=-\left(7+\dfrac{1}{2017}\right)\\ 8S=7+\dfrac{1}{2017}\\ S=\dfrac{7+\dfrac{1}{2017}}{8}\)

Vậy ...