Question

Does the following infinite series converge or diverge? Explain your answer. 1/2 - 2/9 + 4/27 - 8/81

Answer 1

Something tells me the first term is supposed to be
`\frac13`, not
`\frac12`. In that case, the series would be


`\displaystyle\sum_(n=0)^\infty((-2)^n)/(3^(n+1))=\frac13\sum_(n=0)^\infty\left(-\frac23\right)^n`

which is geometric with a ratio less than 1, so the infinite series will converge, and moreover will converge to


`\frac13\cdot\frac1{1+\frac23}=\frac15`

Even if the first term was
`\frac12`, the series will still converge, but we can't write it as we did above. Not counting the first term, we still have a convergent geometric series.