Question

Use the formula to evaluate the series 1/4 + 1/9 + 1/27 + 1/81.

Answer 1

Is the first term supposed to be 1/4, or did you actually mean 1/3?

I'll assume the latter. Write


`S=\frac13+\frac19+\frac1{27}+\frac1{81}`

`S=\frac13+\frac1{3^2}+\frac1{3^3}+\frac1{3^4}`

`\frac13S=\frac1{3^2}+\frac1{3^3}+\frac1{3^4}+\frac1{3^5}`

`\implies S-\frac13S=\frac13-\frac1{3^5}`

`\frac23S=\frac13\left(1-\frac1{3^4}\right)`

`S=\frac12\left(1-\frac1{81}\right)`

`S=(40)/(81)`