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 $\dfrac13$, not $\dfrac12$. In that case, the series would be

$\displaystyle\sum_{n=0}^\infty\frac{(-2)^n}{3^{n+1}}=\frac13\sum_{n=0}^\infty\left(-\frac23\right)^n$

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

$\dfrac13\cdot\dfrac1{1+\frac23}=\dfrac15$

Even if the first term was $\dfrac12$, 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.